Spurs had struggled to break down a disciplined Palace side for much of the game and it looked like they would have to settle for a point.
But Eriksen fired into the bottom corner from 30 yards late on to keep Spurs within four points of Chelsea with five games remaining.
Palace, who lost influential defender Mamadou Sakho to injury in the second half, rarely threatened as they concentrated on frustrating the visitors.
The win means Tottenham move on to 74 points, surpassing their previous best ever Premier League total of 72 - set in 2012-13 - when they finished fifth.
If they beat rivals Arsenal on Sunday it will ensure they finish above the Gunners in the table for the first time since the 1994-95 season.
Palace, meanwhile, remain 12th - seven points above the relegation zone.
Relive Tottenham's win at Crystal Palace
Tottenham needed victory to not just stay in touch with Chelsea but also put behind them the FA Cup semi-final defeat to the Blues on Saturday.
Spurs boss Mauricio Pochettino had been adamant that the 4-2 loss at Wembley would not hurt his players mentally in the title pursuit but that assessment initially looked incorrect as they struggled against a well-drilled Palace.
Eriksen, so often the centre of everything good Tottenham did against Chelsea, was kept quiet while Dele Alli and Harry Kane struggled to provide a spark in attack as the visitors finished the first half with just one shot on target.
Pochettino brought on Son Heung-min and Moussa Sissoko for the second half and changed to a back four in an effort to find a breakthrough. As time went on it looked more and more likely that victory would elude them but to their credit they stayed patient before a moment of magic from Eriksen finally unlocked the Palace defence.
The forward took full advantage of a moment when he was afforded a rare bit of space, shooting from distance beyond the reach of Palace keeper Wayne Hennessey.
It was not a classic performance by a Tottenham side that has scored at least three goals in each of their last three Premier League games but it was one that showed they have the ability to dig in and grind out a result - a side of their game that could prove crucial in the title run in.
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It could have been a different game if Tottenham lost Victor Wanyama to a second booking in the first half.
After picking up an early yellow card for a bad foul, the midfielder slid in late on Andros Townsend. Referee Jon Moss took Wanyama to one side, but let him off with a warning.
Pochettino opted to withdraw Wanyama at half time, but Palace boss Sam Allardyce felt he should never have had the option to do that in the first place.
"Jon Moss should have sent him off," said Allardyce. "The second challenge was probably more of a booking than the first one.
"It is disappointing for us, it is a mistake but at the end of the day we lost a game and we can only think about ourselves right now."
Sakho has been a key player in Palace's upturn in form, with every performance since the defender's January arrival on loan from Liverpool serving only to enhance his transfer value.
Against Tottenham he was once again excellent, keeping the attacking talent of Kane, Alli and Eriksen quiet throughout the first half.
One moment in particular stood out when, under pressure from Kane, he coolly chested the ball down inside his own area before calmly clearing, prompting home fans to chant "sign him up, sign him up".
But they must now face up to the possibility of Sakho being absent from the Palace backline after he suffered what looked like a bad injury early in the second half, falling awkwardly following a challenge.
Allardyce's side are unlikely to go down thanks to an incredible run of just two defeats in their last nine Premier League games but Palace fans will be hopeful of seeing Sakho in a Palace shirt again this season.
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Crystal Palace manager Sam Allardyce: "Outstanding team effort by the players, who have had less time to recover against an exceptionally good side.
"Our application was outstanding and we gave Tottenham a hell of a game in the first half, nip and tuck, but of course it would happen that we would tire given the lack of recovery time compared to Tottenham."
Tottenham boss Mauricio Pochettino: "It was unbelievable. Very good performance. I think second half we played much better than in the first half. It was difficult in the first half for us to move the ball and find the space but we changed the shape at half time and it was more fluid, we started to find the space and started to push Palace deeper and deeper.
"It was good to get the three points and be alive in the race for the title. The challenge is to keep going. It is always better to win but it is true [the Arsenal game] is a big derby, perhaps the last at White Hart Lane and I think it will be an exciting game."
It's the big one for Tottenham on Sunday as they host fierce rivals Arsenal in the Premier League (16:30 BST). Crystal Palace, meanwhile, are at home to Burnley in Saturday's evening kick-off (17:30 BST).
Match ends, Crystal Palace 0, Tottenham Hotspur 1.
Second Half ends, Crystal Palace 0, Tottenham Hotspur 1.
Wilfried Zaha (Crystal Palace) is shown the yellow card.
Moussa Sissoko (Tottenham Hotspur) wins a free kick on the left wing.
Foul by Wilfried Zaha (Crystal Palace).
Substitution, Tottenham Hotspur. Kieran Trippier replaces Dele Alli.
Corner,  Tottenham Hotspur. Conceded by Jeffrey Schlupp.
Corner,  Tottenham Hotspur. Conceded by Yohan Cabaye.
Corner,  Tottenham Hotspur. Conceded by Jeffrey Schlupp.
Ben Davies (Tottenham Hotspur) wins a free kick in the defensive half.
Foul by Andros Townsend (Crystal Palace).
Attempt blocked. Martin Kelly (Crystal Palace) header from the centre of the box is blocked. Assisted by Andros Townsend with a cross.
Corner,  Crystal Palace. Conceded by Ben Davies.
Attempt missed. Toby Alderweireld (Tottenham Hotspur) left footed shot from the centre of the box is close, but misses to the right. Assisted by Ben Davies with a headed pass following a corner.
Corner,  Tottenham Hotspur. Conceded by Damien Delaney.
Corner,  Tottenham Hotspur. Conceded by Damien Delaney.
Corner,  Tottenham Hotspur. Conceded by Jeffrey Schlupp.
Corner,  Tottenham Hotspur. Conceded by Luka Milivojevic.
Substitution, Crystal Palace. Fraizer Campbell replaces Christian Benteke.
Luka Milivojevic (Crystal Palace) is shown the yellow card for a bad foul.
Dele Alli (Tottenham Hotspur) wins a free kick in the defensive half.
Foul by Luka Milivojevic (Crystal Palace).
Goal!  Crystal Palace 0, Tottenham Hotspur 1. Christian Eriksen (Tottenham Hotspur) right footed shot from outside the box to the bottom left corner. Assisted by Harry Kane.
Attempt saved. Harry Kane (Tottenham Hotspur) header from the centre of the box is saved in the centre of the goal. Assisted by Ben Davies with a cross.
Kyle Walker (Tottenham Hotspur) wins a free kick on the right wing.
Foul by Christian Benteke (Crystal Palace).
Foul by Moussa Sissoko (Tottenham Hotspur).
Jason Puncheon (Crystal Palace) wins a free kick in the defensive half.
Attempt missed. Eric Dier (Tottenham Hotspur) header from the right side of the box misses to the right. Assisted by Christian Eriksen with a cross following a corner.
Corner,  Tottenham Hotspur. Conceded by Jeffrey Schlupp.
Christian Eriksen (Tottenham Hotspur) wins a free kick in the defensive half.
Foul by Wilfried Zaha (Crystal Palace).
Foul by Ben Davies (Tottenham Hotspur).
Yohan Cabaye (Crystal Palace) wins a free kick in the defensive half.
Toby Alderweireld (Tottenham Hotspur) wins a free kick on the right wing.
Foul by Yohan Cabaye (Crystal Palace).
Joel Ward (Crystal Palace) is shown the yellow card for a bad foul.
Son Heung-Min (Tottenham Hotspur) wins a free kick on the left wing.
Foul by Joel Ward (Crystal Palace).
Attempt missed. Kyle Walker (Tottenham Hotspur) right footed shot from outside the box is close, but misses to the right. Assisted by Dele Alli.
Tottenham kept up the pressure on Premier League leaders Chelsea as Christian Eriksen's superb long-range strike secured a hard-fought victory at Crystal Palace.