The Scottish Premiership pair played out an eventful tie at Tannadice Park, with three players sent off and two penalties awarded.
A melee started between both sets of players which was only stopped when the linesman intervened as the referee's back was turned.
The hosts continued to apply the pressure and were awarded a free kick in a dangerous position when Aidan Connolly was brought down on the edge of the box by Scott Brown. Nadir Ciftci took the set piece, but he failed to test Gordon as he dragged his strike wide of the post.
Just before the break, both sides hit the post within seconds of each other as Dow could only guide John Rankin's strike onto the woodwork as he tried to beat Gordon.
At the other end, Anthony Stokes was left one-on-one with Cierzniak moments later, and although he beat the keeper, his effort came back off the frame of the goal.
There was further drama to come when Connolly was brought down in the area by Stokes and a penalty was awarded. Ciftci took the spot kick and placed the ball into the bottom corner.
In the second half, Celtic surged forward to get back into the game and were awarded a penalty when Paul Dixon was adjudged to have blocked Johansen's goalbound strike with his hand and was duly sent off.
Leigh Griffiths took the spot kick, but saw his strike saved by Cierzniak, who dived to his left and palmed the ball to safety.
Cierzniak kept United ahead as he made another fine stop, this time to deny John Guidetti from close range as the striker crashed a volley at goal.
However, Celtic got their reward for their pressure in the second half as Johansen lifted a cross to the back post for the run of Griffiths, who guided a header past Cierzniak and into the bottom corner.
The Bhoys were inches away from taking the lead as Nir Biton lined up a strike from distance, but his effort flew just past the post.
Ronny Deila's men pressed again late on but couldn't find the elusive winner as Cierzniak kept them at bay, earning his side a replay.