Man United manager Ole Gunnar Solskjaer made only two changes from the side that won 5-0 in the first-leg in Linz, as Timothy Fosu-Mensah and Jesse Lingard replaced Bruno Fernandes and Luke Shaw in the line-up.
The game was open in the early stages but the visitors came the closest to opening the scoring in the tenth minute after Andres Andrade guided his header onto the crossbar before Phlipp Wiesinger missed the bottom corner by a whisker with a low strike from the follow-up.
LASK were putting United under a lot of pressure at the back but they were not taking their chances and the Red Devils had a chance to punish them just before the half-hour mark after Harry Maguire latched onto a corner, unmarked inside the box but he guided his header wide.
After a slow first half, Solskjaer's charges went behind ten minutes from the break after Wiesinger curled home a first-time strike into the top corner, leaving Sergio Romero with no chance between the sticks.
However, Lingard put United back in the driving seat with the equaliser straight from the restart, as he went clear on goal from Juan Mata's defence-splitting pass and kept his composure to side-foot a low strike past goalkeeper Alexander Schlager.
The Red Devils had their tails up after levelling matters but the Black-Whites almost restored their lead with 20 minutes remaining after Dominik Reiter broke away on the left side of the box and drilled a powerful low strike towards the near post that gave Romero a fright as he blocked the strike with his trailing leg.
The game still had some edge in it despite United cruising on aggregate and Solskjaer made a double change in the last five minutes, as Anthony Martial came on for Daniel James and teenager Teden Mengi replaced Fosu-Mensah, both for fresh legs.
Martial had an immediate impact, as he put the Red Devils ahead with three minutes left after exchanging passes with Mata on the edge of the box to slot through the legs of Alexander Schlager as United claimed a 2-1 win on the night and 7-1 on aggregate.
Shakhtar Donetsk 3-0 Wolfsburg (Shakhtar won 5-1 on agg.)
Copenhagen 3-0 Istanbul Basaksehir (Copenhagen won 3-1 on agg.)
Inter Milan 2-0 Getafe (Inter advance the single-leg tie)