The visitors were closing in on three points thanks to Christian Nade's first-half penalty and a fine finish from Robert Thomson.
But Martin Boyle's late effort - after Daniel Harvie scored an own goal and James Keatings had a penalty saved - earned Hibs a point.
A mistake from goalkeeper Alan Martin handed Boyle the equaliser.
It meant two lost points for Dumbarton, who fall behind Raith Rovers, into eighth spot after the Kirkcaldy side's win over Dundee United.
Dumbarton looked extremely well organised at the back in the first half, with space for the Hibs strikers at a premium.
And the visitors were awarded the chance to take a shock first-half lead when Marvin Bartley needlessly bundled Andy Stirling to the ground just inside the home team's 18-yard box.
Former Hearts striker Christian Nade made no mistake in despatching the ball beyond goalkeeper Ofir Marciano.
When the home side equalised through a Harvie own goal, few inside the ground would have expected anything else other than Hibs going on to secure a victory.
However, Thomson restored the visitors' lead with a stunning shot from 22 yards out to give the travelling support hope their team could secure their first win at Easter Road in more than 30 years.
Martin, on loan from Hamilton Academical, had an afternoon he is likely to remember for a long long time.
He kept Hibs at bay with stunning saves from Keatings and Brian Graham. None better than the penalty stop from Keatings that denied Hibs an equaliser after they laid siege on the visitors' goal.
However, only he knows what he was thinking about when he dropped the ball on the edge of his box, allowing Boyle to nod the ball into the empty net to salvage a point.
If, or indeed when, Neil Lennon manages to get his Hibs side over the line and secure their place back at Scottish football's top table, he will have plenty to think about regarding who he should retain on the playing staff.
With 10 draws in the league this season, the Easter Road side are far from the finished article and the inconsistency shown at times by many of his squad could lead them towards the exit door in the summer.
With two huge games coming up in the next 10 days against their closest challengers, Falkirk and Greenock Morton, the league title could all but be in the bag.
However, any slip ups and those two, under two of the shrewdest managers around, Peter Houston and Jim Duffy, will look to pounce.
Hibernian head coach Neil Lennon: "I'm frustrated as we deserved to win the game. Ridiculous goals we are giving away.
"Our attitude towards the game in the first half wasn't good enough.
"This team has to be more consistent. I'm not scratching my head because I see it so often now.
"We are at home, but the malaise sets in and they are not learning.
"If they think they are going to get to the Premiership playing like that, they won't - and, if they think they are going to play like that in the Premiership, they will have to think again because it's not acceptable."
Dumbarton manager Stephen Aitken: "I certainly enjoyed the performance of the players. We enjoyed the result.
"It could have been a bit better, but if we are assessing the game in an honest manner then Hibs bossed the game in the second half.
"The goalkeeper makes a save from the penalty, so we have to be happy. He is gutted about the equaliser, but at the same time, he kept us in the game.
"He walked in and apologised for costing his team three points, but I said to him he gained us a point."
Match ends, Hibernian 2, Dumbarton 2.
Second Half ends, Hibernian 2, Dumbarton 2.
Foul by Brian Graham (Hibernian).
(Dumbarton) wins a free kick in the defensive half.
Corner,  Hibernian. Conceded by Gregor Buchanan.
Corner,  Hibernian. Conceded by Alan Martin.
Attempt missed. Jordan Forster (Hibernian) right footed shot from the centre of the box is too high.
Corner,  Hibernian. Conceded by Alan Martin.
Attempt saved. Brian Graham (Hibernian) right footed shot from the left side of the box is saved in the bottom right corner.
Attempt missed. Martin Boyle (Hibernian) right footed shot from the centre of the box misses to the right following a corner.
Corner,  Hibernian. Conceded by Daniel Harvie.
Goal!  Hibernian 2, Dumbarton 2. Martin Boyle (Hibernian) right footed shot from very close range to the bottom left corner.
Attempt missed. Dylan McGeouch (Hibernian) right footed shot from outside the box misses to the left.
Attempt saved. John McGinn (Hibernian) right footed shot from outside the box is saved in the top centre of the goal.
Substitution, Dumbarton. Garry Fleming replaces Christian Nade.
Substitution, Hibernian. Brian Graham replaces Andrew Shinnie.
Christian Nade (Dumbarton) is shown the yellow card.
Jordan Forster (Hibernian) is shown the yellow card.
Attempt missed. James Keatings (Hibernian) right footed shot from the left side of the box misses to the right.
Corner,  Hibernian. Conceded by Darren Barr.
Penalty saved! James Keatings (Hibernian) fails to capitalise on this great opportunity,  left footed shot saved  in the bottom left corner.
Corner,  Hibernian. Conceded by Alan Martin.
Gregor Buchanan (Dumbarton) is shown the yellow card.
Penalty Hibernian. Grant Holt draws a foul in the penalty area.
Penalty conceded by Darren Barr (Dumbarton) after a foul in the penalty area.
Substitution, Hibernian. Jordan Forster replaces David Gray.
Corner,  Hibernian. Conceded by Daniel Harvie.
Substitution, Dumbarton. Lewis Vaughan replaces Calum Gallagher.
Corner,  Hibernian. Conceded by Gregor Buchanan.
Attempt saved. Grant Holt (Hibernian) header from the centre of the box is saved in the bottom left corner.
Attempt blocked. David Gray (Hibernian) right footed shot from outside the box is blocked.
Substitution, Hibernian. Dylan McGeouch replaces Marvin Bartley.
Goal!  Hibernian 1, Dumbarton 2. Robert Thomson (Dumbarton) right footed shot from outside the box to the bottom right corner. Assisted by Christian Nade.
Own Goal by Daniel Harvie, Dumbarton.  Hibernian 1, Dumbarton 1.
Attempt missed. David Gray (Hibernian) header from the right side of the box is too high following a corner.
Corner,  Hibernian. Conceded by Alan Martin.
Attempt saved. Andrew Shinnie (Hibernian) right footed shot from outside the box is saved in the bottom right corner.
Robert Thomson (Dumbarton) is shown the yellow card for a bad foul.
Foul by Robert Thomson (Dumbarton).
Marvin Bartley (Hibernian) wins a free kick in the defensive half.
Championship leaders Hibernian twice came from behind to salvage a draw at home to Dumbarton.