Louis van Gaal's men trailed for much of the second half at Upton Park on Sunday before Daley Blind's stoppage-time volley rescued a point from a difficult encounter.
The result meant United remained in fourth place after Southampton had leapfrogged them with a 1-0 win at QPR 24 hours earlier.
"In the end, we are disappointed in the dressing room that we didn't take all three points," defender Jones told MUTV.
"But West Ham pressed us well and I think a draw was a fair result. Ultimately, we are disappointed. We needed to win and it was a big game for us.
"We have not lost, which is a plus, but we could have gained some ground. It was not to be. We have got another big game on Wednesday night now and we need to win that one as well.
"The manager said in the dressing room after the game that he couldn't fault our spirit. If we had shown that more in the first half then we might have come away with a victory.
"But we need to address and evaluate this game. Then we will move on to the Burnley game."
Like United, Burnley were also left disappointed on Sunday as they let slip a two-goal lead against West Brom, who fought back to earn a 2-2 draw at Turf Moor.
It was the second time in successive home league games that Burnley had squandered points after going 2-0 up, hardly ideal preparation for Sean Dyche's men with a difficult set of fixtures to come.
In their next eight games, fourth-bottom Burnley face sides all currently in the top nine, but Dyche remains unfazed by the daunting challenge ahead.
"We know the journey we're on," he said. "We know what it takes and it's another reminder of it [against West Brom].
"Onwards and upwards for us, that's the way we look at it. That's the clarity we need for the players and that's the clarity we ensure they've got."
Stephen Ward is back in training for Burnley after an ankle injury, although Matt Taylor has suffered a setback in his rehabilitation from an Achilles problem.
Dyche has named the same starting XI for 10 league games in a row - two short of the Premier League record - but that run could end after Dean Marney suffered a knee injury on Sunday and did not re-emerge for the second half.
For United - who were held to a goalless draw at Burnley earlier in the season - Luke Shaw is suspended following his late red card at West Ham and Michael Carrick remains out with a muscle rupture.
