The hosts had the lion's share of possession in the opening stages, creating the first real goal-scoring opportunity in the 13th minute.
The Spaniard saved well down low, and then sprung back to his feet to deny Jermain Defoe as he followed up from the rebound.
The linesman had raised his flag for offside, meaning that the England striker's effort would not have counted anyway.
United soaked up some pressure before breaking the deadlock in the 25th minute.
The response from Andre Villas-Boas's men was a positive one, as Defoe tried to find a way past De Gea at his near post, only for the stopper to tip the ball away.
He then saved brilliantly from Gareth Bale to preserve United's lead until the half-time interval.
Spurs began the second half brightly and should have drawn level through Clint Dempsey in the 51st minute.
Defoe and Bale had both wasted decent opportunities, before the American was allowed to meander his way through the defence and shoot from around 12 yards out, only for De Gea to come to United's rescue once again.
The introduction of Wayne Rooney changed the game slightly, as the visitors started to exploit the space left by Spurs' advances down the pitch.
The 27-year-old had a reasonable shout for a penalty when Steven Caulker looked to have taken his leg from underneath him inside the area, but the referee waved play on.
Another big chance came the way of the hosts with 78 minutes on the clock. Defoe found himself in acres of space inside the area, but Rio Ferdinand flung himself in front of the goal-bound shot.
It looked as if United would hold out for three points that would see their seven-point lead at the top of the Premier League restored, but it was not to be.
Lennon latched onto De Gea's punch out and squared for the American, who sent his shot into the corner of the net to snatch a point deep into injury time.