Ander Herrera's first-half strike gave United a lead they deserved in a game low on intensity with both teams already assured of places in next season's Champions League.
Arsenal barely raised a gallop for the first hour but finally came to life in the closing stages and levelled when substitute Walcott's cross was diverted past keeper Victor Valdes off Tyler Blackett.
Former Barcelona keeper Valdes was on for David De Gea, who went off with an injury on what may well be his final Old Trafford appearance as speculation mounts that he will make a summer move back to his native Spain with Real Madrid.
One man who looked like he was definitely saying farewell to Old Trafford was on-loan striker Radamel Falcao, who struggled badly for an hour before being replaced by Robin van Persie, taking time to wave to United's fans as he walked off.
Media playback is not supported on this device
In his programme notes, manager Louis van Gaal suggested a club of United's stature should not be celebrating fourth place - but it was hard to argue with his assertion that this season has been a move in the right direction after the dismal brief reign of his predecessor David Moyes.
Arsenal will confirm a third-placed finish, and a Champions League group-stage spot, with victory against the Black Cats at Emirates Stadium, leaving United to contest the competition's qualifiers.
Van Gaal addressed Old Trafford briefly after the final whistle before the Stretford End chanted loudly for De Gea in a clear attempt to convince him that this was the place where he should stay. Whether it will work is open to question.
Such was the lame nature of Arsenal's first-half performance that they failed to muster a single shot, the first time they have produced this miserable statistic since a 2-1 defeat at Liverpool in November 2004.
United were far more energetic and were rewarded with the lead after 30 minutes when Herrera arrived at the far post to meet Ashley Young's cross after he had escaped from Hector Bellerin.
Van Gaal made a predictable change on the hour, replacing the labouring Falcao with Van Persie. The Colombian's body language suggested that this was the final time he would be seen at Old Trafford as a Manchester United player.
The other United player whose future has become a matter of public debate, De Gea, showed why he is so highly prized when finally called into action just after the hour, reacting quickly to block Olivier Giroud as he threatened from an angle.
Media playback is not supported on this device
He needed Marcos Rojo to rescue him when Arsenal, finally showing signs of life, threatened again. Aaron Ramsey lofted the ball over the onrushing De Gea but Rojo recovered to clear the danger.
De Gea then left the action prematurely as he went down injured and was replaced by Valdes.
The debutant had barely had time to make any sort of contribution when Arsenal equalised, Walcott's cross taking a big deflection off Blackett to wrong-foot the keeper.
Arsenal will secure third place in the Premier League if they beat Sunderland on Wednesday after Theo Walcott's deflected cross gave them a late equaliser and a point at Manchester United.