Both teams had a man sent off - the Terrors' Lewish Toshney, and solitary goalscorer Jason Cummings who picked up two yellow cards apiece at Tannadice.
Cummings volleyed in the only goal after 38 minutes, shortly after Toshney's dismissal.
But his 75th-minute sending off for punching the ball into the United net made for a nervous finale for Hibs.
Neil Lennon's side extended their lead at the Championship summit to nine points, ahead of second placed Falkirk, who play Ayr United on Saturday. Ray McKinnon's United are 10 points adrift in third place, having played a game fewer than Hibs.
Clearly, the home side had to start on the front foot and pressure their opponents. With games running out they really had to win to close the gap the Easter Road side.
United's eagerness to play at pace and be in Hibs' faces ultimately cost them, epitomised by Toshney's rashness.
The right-back was lucky to escape one early reckless challenge, but was correctly punished for another on Andrew Shinnie, and when he clattered the midfielder once again, it was a simple decision for referee Don Robertson.
Hibs quickly asserted themselves by going in front and it became too big a task despite Cummings' offer of assistance.
United didn't lack fight. They lacked composure and quality on the night. A first home defeat this season was badly timed.
They're not entirely out of the title race, but the smart money would be on a play-off place for the Tannadice side.
Head coach Lennon might not admit it but few will back anyone other than Hibernian to take the Championship title now.
Cummings' fabulous first-half finish was timely, not too long after Toshney's red card.
The striker stole in behind a defence that seemed intent on playing a high line. Shinnie exposed that with a lovely lofted ball into Cummings' path and he made no mistake with a vital, instinctive goal.
Cummings was also unlucky to be flagged offside when clean through and might have earned a penalty when William Edjenguele appeared to catch him near the goalline just before half-time.
It was the 21-year-old's 20th goal of the season - great work from him all-round, until later.
Martin Boyle should have ended the contest when he had time, and space six yards out, but volleyed wildly over.
The impressive Shinnie thought he had sealed the win after great work from Boyle, only for Cammy Bell to parry his shot over the bar.
Hibs rarely looked troubled with goalkeeper Ofir Marciano a virtual spectator.
This was a massive step for Lennon's men, and with three home games to follow, they are very much on track for the top flight.
For all his positives, Cummings risked this win with few minutes of madness. He was unnecessarily booked for throwing the ball away when ruled offside.
Minutes later he threw himself at a cross, blatantly punched the ball in the net and was correctly ordered off.
It was utter madness and offered United a lifeline from nowhere. Luckily for him, his team mates saw it through.
In total, 10 players were booked, and Lennon was ordered to the stand late on. However, Hibs got the job done, and deserved to finish with three valuable points.
Dundee United manager Ray McKinnon: "It was extremely difficult when we went down to 10 men.
"I thought we started the game excellently and controlled the first 15 minutes, but the sending-off certainly had a massive impact on the game.
"I haven't seen it again, but Lewis has gone for the ball and then he has pulled out. It was clear for everybody on our bench.
"The ref is entitled to make that decision, but we were very disappointed. You have to look at all the decisions that were made and some of inconsistencies and tackles that went unpunished.
"But what is important is what we do from now until the end of the season and focus on winning every game.
"I thought we showed great character to battle back hard with 10 men. We will take the positives out of that and fight for every point until the end and see where we end up."
Hibernian head coach Neil Lennon: "Jason's got to learn to control his temper.
"I've been there - I got many a telling off from Martin O'Neill in my younger days - but he's an outstanding talent and he's come up with a really big goal in a big game and he does it consistently and it's not a coincidence.
"We were outstanding. The only disappointment was the sending off really, which put us on the back foot for 10 minutes.
"The rest of the game, we played really strongly and we looked a very good side. Thoroughly deserved to win.
"I wanted them to put down a marker for the rest of the season and hopefully they've done that.
"It is a difficult place to come and win, but I thought we were head and shoulders above Dundee United.
"I underplayed it this week, but it is probably the biggest game we've had league wise for two or three seasons and they've responded brilliantly."
Match ends, Dundee United 0, Hibernian 1.
Second Half ends, Dundee United 0, Hibernian 1.
Foul by John McGinn (Hibernian).
Paul Dixon (Dundee United) wins a free kick on the right wing.
Brian Graham (Hibernian) wins a free kick in the defensive half.
Foul by Willo Flood (Dundee United).
Hand ball by Willo Flood (Dundee United).
Jordan Forster (Hibernian) is shown the yellow card for a bad foul.
Foul by Jordan Forster (Hibernian).
Mark Durnan (Dundee United) wins a free kick on the right wing.
Substitution, Hibernian. Jordan Forster replaces Martin Boyle.
Foul by John McGinn (Hibernian).
Frank van der Struijk (Dundee United) wins a free kick in the attacking half.
Attempt blocked. Paul Dixon (Dundee United) right footed shot from outside the box is blocked.
John McGinn (Hibernian) is shown the yellow card for a bad foul.
Foul by John McGinn (Hibernian).
Simon Murray (Dundee United) wins a free kick in the defensive half.
Darren McGregor (Hibernian) wins a free kick in the attacking half.
Foul by Mark Durnan (Dundee United).
Substitution, Dundee United. Nick van der Velden replaces Scott Fraser.
Lewis Stevenson (Hibernian) wins a free kick on the right wing.
Foul by Thomas Mikkelsen (Dundee United).
Attempt saved. Andrew Shinnie (Hibernian) left footed shot from the left side of the six yard box is saved in the bottom left corner.
Corner,  Hibernian. Conceded by Cammy Bell.
David Gray (Hibernian) wins a free kick on the left wing.
Foul by Thomas Mikkelsen (Dundee United).
Substitution, Hibernian. Brian Graham replaces Grant Holt.
Attempt missed. David Gray (Hibernian) header from the centre of the box misses to the right.
Attempt blocked. Grant Holt (Hibernian) right footed shot from the right side of the box is blocked.
Corner,  Hibernian. Conceded by Mark Durnan.
Second yellow card to Jason Cummings (Hibernian) for hand ball.
Hand ball by Jason Cummings (Hibernian).
Attempt blocked. Jason Cummings (Hibernian) left footed shot from outside the box is blocked.
Jason Cummings (Hibernian) is shown the yellow card.
Delay over. They are ready to continue.
Delay in match Scott Fraser (Dundee United) because of an injury.
Delay in match David Gray (Hibernian) because of an injury.
Attempt blocked. Jason Cummings (Hibernian) right footed shot from the centre of the box is blocked.
Corner,  Hibernian. Conceded by Frank van der Struijk.
Attempt blocked. Andrew Shinnie (Hibernian) right footed shot from the centre of the box is blocked.
Scottish Championship leaders Hibernian moved 10 points clear of title rivals Dundee United with a fractious victory.