Frank Lampard had a chance to break the deadlock with almost half an hour gone, but the midfielder scuffed his shot, giving the United keeper an east stop.
The home side's best chance of the opening half came just before the interval, as Ryan Giggs slid the ball in for Robin van Persie, but the league's leading goalscorer could not keep the ball on target.
© PA Photos
Rafael Benitez's side dominated possession at the start of the second half, but the team failed to test Lindegaard.
Manager Sir Alex Ferguson brought Wayne Rooney on for the final 20 minutes as the champions pushed for a winner, and the striker almost found the back of the net as he attempted to head in from an Antonio Valencia cross, but he could not keep the effort under the bar.
The visitors managed to find a goal in the 86th minute with Mata picking the ball up inside the box and deflecting it in off Phil Jones.
As Rafael charged down the wing looking for an quick equaliser, the Brazilian tangled with Luiz, resulting in a red card for the United defender and a scuffle between a number of players from both sides.
With a one-man advantage, Chelsea were able to protect their advantage through stoppage time and secure the win.
The result moves Chelsea above Arsenal into third in the Premier League table.