Giggs struck with just seven minutes remaining, with his quickly-taken effort sparking furious protests from the home team, some of whom threatened to leave the pitch.
A largely forgettable first half offered few clues to the drama that would ensue later on in the evening, with neither side able to forge a clear opening until just before the interval, when home keeper Tony Sylva denied Henrik Larsson and Cristiano Ronaldo.
United started the second half in much brighter fashion, with Wayne Rooney conjuring up some magic six minutes in, skipping away from three defenders before trying to find Giggs at the far post, but the pass was just too strong.
Soon after Ronaldo broke clear of the home defence, but his shot was straight at Sylva when it looked easier to score.
Just before the hour Lille almost made United pay, as the returning Edwin Van der Sar stuck out a foot to block a close-range effort from Mathieu Bodmer.
A few minutes later Lille thought they had the breakthrough, but Peter Odemwingie's point-blank header was ruled out for a push on Nemanja Vidic.
Sir Alex Ferguson responded by brining on Louis Saha and he had a chance to score with his first touch, but he miskicked when a long throw found him unmarked in the area.
Odemwingie then curled a left-foot shot narrowly wide of the far post, before Larsson almost broke the deadlock with a superb piece of skill.
Rooney got the ball in from the right, Sylva flapped at the cross and Larsson found space, but lifted his shot just over the far post from a tight angle.
Substitute Johan Audel reached Mathieu Debuchy's near-post cross nine minutes from time but could only direct his shot over, with Van der Sar rooted to the spot.
That left Giggs to nip in for a cheeky matchwinner, with the protests of the home players and fans lasting all the way to the final whistle, but United left with a valuable advantage for the Old Trafford return.