With Southampton grabbing a late winner at Queens Park Rangers and Tottenham Hotspur reviving their Champions League charge with a win over rivals Arsenal on Saturday, United needed all three points to consolidate their place in the top four.
However, West Ham dominated the early stages at Upton Park and Cheikhou Kouyate's neat second-half volley looked to have given the Londoners the victory only for Blind to fire home from the edge of the area after a poor clearance.
A draw for United keeps them fourth on 44 points from 24 matches, one behind Southampton and one ahead of Spurs, while West Ham remain eighth with 37 points.
Chelsea lead the way with 56 points after winning 2-1 at Aston Villa on Saturday, seven ahead of Manchester City who were held 1-1 at home by struggling Hull City.
"We played very badly in the first half and showed a lot of spirit in the second half," United manager Louis van Gaal told Sky Sports.
"We gave away that goal and then played football but you have to do it from the first minute. But what spirit after the goal and I thanked my players for that."
Earlier, West Bromwich Albion came from two goals behind to draw 2-2 at fellow strugglers Burnley and claim what could prove to be a crucial point in the fight for Premier League survival.
In a mid-table battle at St James' Park, Peter Crouch's late equaliser earned Stoke City a 1-1 draw against Newcastle United and kept Mark Hughes's side in the top 10.
MOUNTING PRESSURE
West Ham made the better start at Upton Park and Enner Valencia forced two smart saves from David De Gea as the home side looked to expose the defensive frailties that have hindered United this season.
James Tomkins was the next West Ham player to test De Gea when the defender's header from a free kick called the Spanish keeper into action.
But Kouyate finally fired the Hammers in front after the break when he neatly juggled the ball in the box before twisting his body and volleying past the stranded De Gea.
United were struggling to create chances but striker Radamel Falcao squandered an opportunity when through on goal before Robin van Persie had a shot well saved by Adrian.
However, United showed their old battling qualities as West Ham failed to clear Marcos Rojo's ball forward and Blind struck to deal the Hammers a blow in added time.
United were forced to play the final few minutes with 10 men when defender Luke Shaw received a second yellow card.
EARLY SPARK
Burnley's Ashley Barnes and Danny Ings provided the early spark at a wet and gloomy Turf Moor as they each scored first half headers to give the hosts a 2-0 lead.
But Chris Brunt earned West Brom a lifeline towards the end of the first half when he nodded in from a corner and substitute Brown Ideye completed the comeback for Tony Pulis's side in the second period when he powered in a header from Brunt's corner.
West Brom are 15th with 23 points from 24 matches, while Burnley are a point above the relegation zone in 17th with 21.
After a drab opening half at St James' Park, Jack Colback fired Newcastle in front minutes after being fortunate not to receive a second yellow card for a foul on Victor Moses.
Gabriel Obertan wasted a chance to double Newcastle's lead and former England striker Crouch punished them in the 90th minute when he headed in Geoff Cameron's pinpoint cross.
Stoke remain 10th with 33 points, while Newcastle are two points behind in 11th spot.
