Falkirk's 2-2 home draw with Queen of the South gave the Tangerines the chance to move above them, and Tony Andreu's header put the hosts ahead.
But Robbie Thomson nodded The Sons level before half-time and Lewis Vaughan's neat finish made it 2-1.
Thomas Mikkelsen tapped home to level but United couldn't find a winner.
They remain in third place, a point behind Falkirk, who travel to face Dumbarton next Saturday, while United complete their regular-season fixtures away to fourth-placed Greenock Morton.
Ray McKinnon's side must win at Morton and hope Falkirk drop points, or United will be consigned to an extra two-legged tie - against Morton - in the play-offs.
The Sons, meanwhile, have all but sealed their place in the Championship for another year after moving three points clear of Raith Rovers in the relegation play-off spot, but with a superior goal difference of nine.
Vaughan, who scored his fourth goal since joining Dumbarton on loan from Raith in January, finds himself in the tricky position of possibly sending his parent club into the relegation play-off.
"Obviously I am contracted to Raith next season and it's a bit of a strange one, as I think Rovers are far too big a club to go down," he told BBC Alba.
"But I am a professional footballer at the end of the day and I am just doing my job. I don't know if I could play for Raith if they end up in the play-off, but I would love to go back there and keep them in the Championship."
Rovers could still avoid the play-off if they beat bottom side Ayr United in Kirkcaldy, and St Mirren lose at champions Hibernian in their final game.
Match ends, Dundee United 2, Dumbarton 2.
Second Half ends, Dundee United 2, Dumbarton 2.
Corner,  Dundee United. Conceded by Daniel Harvie.
Substitution, Dumbarton. David Smith replaces Samuel Stanton.
Attempt missed. Thomas Mikkelsen (Dundee United) header from the centre of the box is just a bit too high.
Corner,  Dundee United. Conceded by Andy Stirling.
Attempt blocked. Tony Andreu (Dundee United) right footed shot from outside the box is blocked.
Blair Spittal (Dundee United) wins a free kick on the right wing.
Foul by Ross McCrorie (Dumbarton).
Attempt blocked. Samuel Stanton (Dumbarton) left footed shot from outside the box is blocked.
Substitution, Dumbarton. Garry Fleming replaces Christian Nade.
Foul by Willo Flood (Dundee United).
Mark Docherty (Dumbarton) wins a free kick in the attacking half.
Attempt missed. Simon Murray (Dundee United) left footed shot from outside the box is too high.
Goal!  Dundee United 2, Dumbarton 2. Thomas Mikkelsen (Dundee United) right footed shot from very close range to the bottom right corner.
Attempt saved. Alistair Coote (Dundee United) right footed shot from outside the box is saved in the centre of the goal.
Attempt saved. Andy Stirling (Dumbarton) right footed shot from the centre of the box is saved in the centre of the goal.
Substitution, Dumbarton. Mark Docherty replaces Lewis Vaughan.
Thomas Mikkelsen (Dundee United) wins a free kick in the defensive half.
Foul by Ross McCrorie (Dumbarton).
Attempt saved. Samuel Stanton (Dumbarton) right footed shot from outside the box is saved in the centre of the goal.
Substitution, Dundee United. Alistair Coote replaces Wato Kuate.
Attempt missed. Simon Murray (Dundee United) left footed shot from the centre of the box is too high.
Attempt missed. Coll Donaldson (Dundee United) right footed shot from the centre of the box is close, but misses to the left.
Corner,  Dundee United. Conceded by Darren Barr.
Attempt saved. Simon Murray (Dundee United) right footed shot from the centre of the box is saved in the bottom right corner.
Attempt missed. Willo Flood (Dundee United) right footed shot from outside the box is too high.
Attempt missed. Wato Kuate (Dundee United) right footed shot from outside the box is too high.
Goal!  Dundee United 1, Dumbarton 2. Lewis Vaughan (Dumbarton) right footed shot from very close range to the bottom left corner. Assisted by Samuel Stanton.
Mark Durnan (Dundee United) wins a free kick in the attacking half.
Foul by Christian Nade (Dumbarton).
Attempt missed. Wato Kuate (Dundee United) right footed shot from outside the box is too high.
Attempt saved. Willo Flood (Dundee United) right footed shot from outside the box is saved in the centre of the goal.
Coll Donaldson (Dundee United) wins a free kick in the defensive half.
Foul by Christian Nade (Dumbarton).
Foul by Christian Nade (Dumbarton).
Wato Kuate (Dundee United) wins a free kick in the attacking half.
Attempt missed. Simon Murray (Dundee United) right footed shot from the left side of the box is close, but misses to the right.
Attempt missed. Andy Stirling (Dumbarton) left footed shot from outside the box is close, but misses to the left.
Christian Nade (Dumbarton) wins a free kick in the defensive half.
Dundee United's hopes of finishing second in the Championship suffered a blow as they could only draw with lowly Dumbarton at Tannadice.