Ralf Rangnick expressed his frustration with United's finishing after a 1-1 draw at Burnley last time out, but he could have few complaints when Jadon Sancho put his side ahead after 21 minutes at Old Trafford.
But Che Adam restored parity just three minutes into the second half and United, which saw two goals ruled out for offside, could not find a winner against Ralph Hasenhuttl's resolute defence.
The draw means United jumps up to fifth, overtaking Arsenal, which is a point behind and has two games in hand, while moving level on points with fourth-placed West Ham United, although the Hammers are in action at Leicester City on Monday (AEDT).
Cristiano Ronaldo seemed set to open the scoring after just six minutes but his effort was cleared off the line by Romain Perraud, before Adams inexplicably headed away from goal after Kyle Walker-Peters teed up a golden opportunity.
United struck first when Sancho poked home from close range after Marcus Rashford drilled across from the right after an incisive Bruno Fernandes pass.
Stuart Armstrong went close in response as his acrobatic attempt tested David de Gea, while Paul Pogba's finish was chalked off for offside against Ronaldo in a frantic first half.
Adams atoned for his earlier miss swiftly after the interval as he equalised with a low finish off the inside of the post after Mohamed Elyounoussi's throughball, before Fraser Forster beat away from Diogo Dalot to keep the Saints level.
Ronaldo thought he had restored United's lead with 19 minutes remaining, but his header was ruled out for offside after Sancho's flick from Fernandes's free-kick, and Forster expertly parried away Harry Maguire's stoppage-time header to earn Southampton a point.