Wayne Rooney twice leveled for the visitors in an action-packed encounter, his second goal coming from the penalty spot on 57 minutes.
Kyle Walker, who was at fault for Rooney's first, and Sandro had put Tottenham ahead, the latter with a spectacular long-range strike.
Having been thrashed 6-0 at Manchester City in their last league match, Andre Villas-Boas's men can take heart from a much more competitive showing and their first league goals since October, while United are now unbeaten in 12 matches across all competitions.
However, both sides remain outside the top four and some way adrift of Arsenal, Tottenham trailing the front-runners by 10 points and the Red Devils by nine.
Villas-Boas made three changes to the side humbled at the Etihad Stadium, with Younes Kaboul, Lewis Holtby and Erik Lamela – who did not even make the bench – omitted. Robin van Persie remained sidelined for the visitors, whose team included four alterations from the 2-2 draw at Cardiff City, with 40-year-old Ryan Giggs among those to miss out.
Rooney saw a goalbound strike superbly blocked by Sandro prior to Walker opening the scoring in the 18th minute. Jonny Evans upended Paulinho on the edge of the area and Tottenham's right-back drilled the resulting free kick under United's jumping wall.
The influential Rooney was forced into a block of his own to deny Michael Dawson before Tottenham – a constant threat on the counterattack – spurned two chances in quick succession either side of the half-hour mark.
Roberto Soldado fired high and wide following a delightful exchange of passes with Paulinho, while Aaron Lennon's left-footed effort forced David de Gea into a save with his feet.
Tottenham were swiftly punished for not extending their advantage in the 32nd minute as Walker diverted a right-wing cross from Phil Jones into the path of Rooney, who fired home from close range.
The home side regained the lead in stunning fashion nine minutes after the break, though, with Sandro cutting inside Tom Cleverley before unleashing a thunderous strike into the top-right corner.
Yet United were level again within three minutes, Rooney firing a penalty straight down the middle for his eighth Premier League goal of the season after Danny Welbeck had been brought down by Hugo Lloris.
Rooney threatened again from a Patrice Evra cross, while Walker put another free kick narrowly wide before providing a delightful cross with which substitute Andros Townsend came agonisingly close to connecting, yet neither team could find a winner.