Bryan Mbeumo put United in front against the run of play 17 minutes from time, but Rodrigo Bentancur was shown a straight red card for a headbutt on Luke Shaw. United failed to make the man advantage count, and Solanke stooped to head in Sandro Tonali's cross in stoppage time to level the score.
Tottenham had dominated much of the match, with Solanke forcing a save from Senne Lammens and Mohammed Kudus blazing wide. Roberto De Zerbi's side took until the closing stages of their fifth league game of the season to score their first goal, and a much-improved performance was undone by a familiar lack of cutting edge up front.
United had barely threatened Antonin Kinsky's goal until Mbeumo was afforded too much room to spin onto Matheus Cunha's pass and fire into the far corner. Carrick's triple substitution on the hour mark, including the introduction of Carlos Baleba after his 70-million-pound ($92.6m) move from Brighton, helped turn the tide.
However, not for the first time this season, United failed to make a man advantage count. Manchester City won 1-0 at Old Trafford last month despite playing for more than an hour with 10 men.
The point edges Spurs up to 18th, with only goals scored keeping them inside the relegation zone. United are just three points better off in 12th.