Marcos Rojo gave United an early lead at Old Trafford, but United academy graduate Josh King equalised from the penalty spot.
Controversy reigned as Tyrone Mings and Zlatan Ibrahimovic both avoided red cards just before the interval - the former for a stamp on the striker; the latter for elbowing the defender in retaliation - as Bournemouth midfielder Andrew Surman then received a second bookable offence for pushing over Ibrahimovic.
The Swedish striker was lucky to be on the pitch, but then missed a second-half penalty as Jose Mourinho's side failed to break down Eddie Howe's Cherries.
Premier League table
The result was a boost for both Arsenal and Liverpool as they face off at Anfield later on Saturday afternoon, with third-placed Tottenham hosting Everton on Sunday before leaders Chelsea travel to West Ham on Monday night.
Crystal Palace secured a vital 2-0 victory against West Brom at The Hawthorns to move three points clear of the relegation zone.
Goals from Wilfried Zaha and Andros Townsend were vital for the Eagles with relegation rivals Swansea and Leicester also winning on Saturday.