However, Peter Houston's men missed two excellent first-half chances and remain firmly in the qualifying tie.
A late goal from substitute Johan Voskamp, a big-money buy from Sparta Rotterdam, gave the Poles victory and inflicted a cruel blow on United.
Houston's side spent much of the match chasing the ball but had two great chances to take the lead through Johnny Russell and David Goodwillie before Slask struck with 15 minutes remaining.
Although they did not get the away goal they wanted, United will take plenty of heart from their performance and on the evidence of this 90 minutes still stand a real chance of going through to the third qualifying round.
United, who were minus the injured Garry Kenneth, Scott Robertson and Danny Swanson, named striker Jon Daly in central defence alongside Sean Dillon, with Scott Severin pushed up into midfield.
United were making their second visit to the Oporowska Stadium, having secured a goalless draw there in the first round of the UEFA Cup in 1980 before winning 7-2 at Tannadice.
They found themselves up against the same manager, with Orest Lenczyk enjoying a second spell at the club, having led them to runners-up place in the Polish league last season.
The Tangerines were virtual spectators for the opening few minutes as Slask showed their quality by passing the ball around almost effortlessly.
And they should have scored after eight minutes when Cristian Diaz cut inside from the left but he pulled his shot wide from 16 yards.
United, the seeded team in the tie, were set up to break on the counter-attack and it paid off just two minutes later when Russell made space for himself on the wing and got past his marker.
The Scotland Under-21 international took the ball round goalkeeper Marian Kelemen and looked certain to score but scuffed his shot wide.
Slask continued to show their football skills, keeping possession for long spells, but United's defence was impressively dogged in keeping them at bay.
And it was United who went closest again through John Rankin, whose shot from the edge of the penalty area needed two attempts from goalkeeper Kelemen to finally get it under his control.
Goalkeepers came to the fore as the half wore on, with United's Dusan Pernis denying Marek Gancarczyk from the edge of the area.
United then passed up the best chance of the game. A flick from Russell carved Slask wide open and set lone striker Goodwillie through one-on-one with the goalkeeper.
The Rangers, Cardiff and Blackpool transfer target looked certain to score but he tried to go round the keeper and was blocked out.
Then it was the turn of Pernis to shine and ensure United went in level at the break. Diaz cut in from the left and clipped a shot towards the far corner which looked like trickling in off the post.
But Pernis threw himself across his goal to get a hand to the ball and turn it away.
The Slovak saved United again on 58 minutes by blocking a Amir Spahic header from Sebastian Mila's cross.
Slask introduced new signing Voskamp on 72 minutes and that proved to be crucial.
The striker soon got on the end of Tadeusz Socha's cross from the right and slotted the ball into the bottom corner from eight yards.
United lost Severin to injury in the 88th minute but will still be quietly confident of overturning their deficit at Tannadice next Thursday night.