Wayne Rooney had a chance on 15 minutes, but his strike was blocked superbly by Sandro.
Spurs went 1-0 up on 18 minutes through Kyle Walker, who blasted a free kick underneath the United wall, giving De Gea no chance.
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The home side enjoyed a spell of dominance following their goal and had two penalty appeals turned down in quick succession on 21 minutes.
Roberto Soldado had a huge chance to double Spurs' lead on 29 minutes, though he could only fire his shot over the bar after a quick counter-attack.
United equalised on 31 minutes against the run of play when a Phil Jones cross rebounded off Walker into the path of Rooney, who poked home from close range to make it 1-1.