Callum Wilson gave the Cherries a deserved opener after a blistering start at the Vitality Stadium.
Eddie Howes side had more opportunities to extend their lead but Anthony Martial equalised before the break – and against the run of play – when he converted Alexis Sanchezs cross.
United, though, emerged far better after the break but the Red Devils surely felt they had squandered their chances of all three points after Rashfords goal-bound effort was miraculously blocked by Nathan Ake.
But United kept piling forward and Rashford got a perfect chance to make amends when Paul Pogba clipped a good ball to the back post and the 21-year-old pounced to hammer home.
Bournemouth will be left ruing their missed chances in the first half and for not capitalising on their dominance in the opening half an hour.
Ryan Fraser and David Brooks started in behind Wilson in a fluid front three and their runs behind Uniteds defence were causing the visitors all sorts of problems in the first half.
Junior Stanislas missed a good opportunity to put Bournemouth ahead, while Fraser tested David De Gea from close range.
Wilsons opener eventually came when United went sleeping from a corner. Juan Mata allowed Stanislas to slip behind him and the midfielder ran onto Lewis Cooks brilliant pass to set-up Wilson for a simple finish.
Uniteds equaliser came completely against the run of play and the visitors had man-of-the-moment Anthony Martial to thank.
Alexis Sanchez, starting in place of Romelu Lukaku who was absent with injury, played a neat cutback and the Frenchman finished into the bottom corner with his left foot.
United were happy to be on level terms at the break and they emerged after the interval as a different team.
Luke Shaw should have put Mourinhos men ahead when he played a brilliant one-two with Martial but his effort was saved by Asmir Begovic, while Sanchez also went close with a header.
United thought theyd missed their opportunity to grab all three points when Ashley Youngs free-kick that hit the crossbar fell to Rashford. The forward, who came on for Sanchez, saw his effort blocked by Ake and Pogbas follow-up was cleared off the line.
A draw would have been a fair result but United needed a win with Manchester City away to come next week and Rashford reedemed himself by smashing home in the 93rd minute after great work from Pogba.