United had their chances to win, but they were few and far between in a desperately poor European Champions League Group D encounter.
Scholes received the first of two yellow cards in the 29th minute when he clattered into Mathieu Bodmer.
Then, in the 63rd minute, the former England man brought down Jean Makoun and was dismissed by Italian referee Stefano Farina for a second bookable offence.
Giggs suffered his injury during the second half when he was caught by an elbow in the face, but the Welshman struggled on until eight minutes from time when he was replaced by Park Ji-Sung.
The Red Devisl, without suspended talisman Wayne Rooney, lacked ideas and found the going tough against Lille's well organised defence.
Two of their best chances came within a minute of each other early on.
In the 16th minute, Phil Bardsley's ball into the Lille box was well laid off by Rio Ferdinand to Scholes, whose blistering shot from the left side of the penalty area was brilliantly saved by goalkeeper Tony Sylva.
Seconds later, Bardsley's low cross from the right found Ruud van Nistelrooy free 12 yards out, but his weak shot was easily saved by Sylva.
Giggs, captaining United on his official 100th European appearance for the club, hit the Lille post after 33 minutes with a curling left-foot free-kick 25 yards out.
But, as the game went on, United found themselves more and more chasing down blind alleys in a bid to break the deadlock.
Their best chance of the second half came in the 65th minute when Giggs fed Cristiano Ronaldo on the right and he cut into the box before hitting a fierce shot which was superbly saved by Sylva.
Lille had a 10th minute appeal for a penalty turn down when Peter Odemwingie went down after going into United goalkeeper Edwin Van der Sar, but Mr Farina rightly waved it away.