Juan Mata went closest to breaking the deadlock in the opening 45 minutes with a free-kick that hit the post and was hacked away on the line.
Ashley Young also saw a shot brilliantly tipped over the bar by Hugo Lloris while Robin van Persie was unable to finish from a few yards out.
United also had chances in the second half as Van Persie missed the target with a volley and Mata blasted over the bar from a promising position.
Spurs rarely threatened at the other end but did finish stronger and could have snatched the points as Ryan Mason fired over with just David de Gea to beat.
Had Mason’s shot hit the back of the net then United would have certainly rued the chances they missed in a largely one-sided first half.
Having named an unchanged Premier League team for the first time since November 2012, United looked far more settled than Spurs – who made four changes – and created all the best openings.
The first sight of goal for the visitors came in the seventh minute when Mata clipped a pin-point ball over the top of the defence for Wayne Rooney, but his poor touch prevented a one-on-one chance.
Mata set up another chance shortly afterwards as he slid a ball through for Radamel Falcao, who couldn’t quite sort his feet out and ended up shooting at Lloris.
Spurs offered little at the other end aside from the odd counter attack and were fortunate not to fall behind after 22 minutes as Mata’s free-kick bounced off the post and was then hacked clear from under the bar.
Chances continued to come and go for United with Van Persie unable to fire home from close range and Rooney seeing a header saved by Lloris.
The Spurs keeper then pulled out a top-drawer save as the half came to a close, tipping Young’s curling shot over the bar.
Even though the hosts improved after the break, United still had opportunities, with Van Persie missing the target with a volley from 10 yards out.
Mata then fired over the bar from Rafael’s pull-back, but United’s grip on the game loosened in the final 20 minutes as Spurs started to look stronger.
De Gea was called upon to save Fazio’s low shot before the home side created their best chance of the match in the 82nd minute.
Harry Kane did brilliantly to hold the ball up against four United defenders and then slid a clever reverse pass through for Mason, who fired wastefully over the bar as he was bearing down on goal.
Spurs continued to press in the final 10 minutes but couldn’t grab another late winner as both sides settled for a share of the spoils less than 48 hours after their last outings.
Tottenham Hotspur: Lloris (7), Chiriches (5), Davies (6), Fazio (6), Vertonghen (6), Chadli (5), Eriksen (6), Mason (6), Stambouli (6), Townsend (5), Kane (6).
Subs: Dembele (5), Lamela (5).
Manchester United: De Gea (6), Evans (5), Jones (6), McNair (6), Carrick (6), Mata (7), Rooney (6), Valencia (5), Young (7), Falcao (6), Van Persie (5).
Subs: Rafael (6), Shaw (5), Smalling (5).
Man of the match: Hugo Lloris.