Goals from Keith Watson and David Goodwillie had overturned Slask's 1-0 first-leg advantage inside five minutes at Tannadice on Thursday night, only for United to quickly concede from a corner as Rok Elsner headed home.
Jon Daly's penalty put Dundee United back in the ascendancy just before the break and the Poles looked to have run out of ideas.
But, with the imminent danger not obvious, Sebastian Dudek struck from 30 yards in the 74th minute and United were unable to mount a serious late charge as around 2,000 Polish fans celebrated an 3-3 aggregate draw and a place in round three.
United were ultimately made to pay for bad misses by David Goodwillie and Johnny Russell in Poland last week, which came before Johan Voskamp's late winner.
Peter Houston had been encouraged by that first-leg display, and Jon Daly's success in the air when he ventured from his unfamiliar role at the back in Poland.
And the return of Garry Kenneth from a knee injury meant the Irishman started up front as Houston reverted to a 4-4-2 formation.
The Poles in the 11,306 crowd, most of them Scots-based, helped create a vibrant atmosphere inside Tannadice but they were briefly silenced with less than two minutes gone.
Willo Flood's left-wing corner went beyond the crowd of bodies and towards the far edge of the box. Scott Severin slid in to win the ball and set up Watson, who rifled the ball just inside Marian Kelemen's near post from 16 yards.
Houston's faith was soon vindicated further when Daly flicked on Sean Dillon's forward ball and set Goodwillie free. The striker cut off the last man as he charged inside from the left and slipped the ball just inside the far corner.
Visiting manager Orest Lenczyk might have feared a repeat of his previous visit to Tannadice at this stage - Slask lost 7-2 in a 1980 UEFA Cup tie during his first spell in charge.
But his side settled quickly and claimed a simple reply when Elsner headed home Sebastian Mila's corner from six yards.
The Poles knocked the ball around confidently and Dudek came close with a great strike from 25 yards, but they looked vulnerable from set-pieces.
Daly came close with two headers from corners by Flood, the second tipped over by the erratic Kelemen.
The goalkeeper looked less than assured when punching away another Flood corner and then came 12 yards off his line for a long throw which Kenneth won in the air. However, Elsner comfortably cleared John Rankin's hooked effort off the line.
Kenneth had a header tipped over and then looked to have been held when attacking another Flood delivery but got away with a risky challenge on Voskamp in the other penalty box.
Kelemen's reckless approach cost his side dear in the 44th minute. The goalkeeper charged off his line and brought down Goodwillie just inside the left edge of the box as the striker chased Dusan Pernis' long kick, and conceded a penalty and yellow card.
Daly confidently converted the penalty into the top corner to put United back on course for the third qualifying round, and almost added another in injury-time when he headed inches past the far post following a high ball.
The second half proved a much cagier affair with United containing the Poles well for long spells.
The hosts eventually began to create chances on the break with Goodwillie and Russell shooting wide from outside the box.
But, with the Poles seemingly running out of ideas, they restored their advantage when Dudek earned himself some space 30 yards out and let fly with a shot that Pernis could only palm into the top corner.
United pushed men forward but their only real chance came in injury-time when Daly shot over from 14 yards as he struggled to keep his balance under pressure.