The Red Devils fell behind after 32 minutes when Jordan Ayew but appeared to have rescued at least a point when James levelled the score in the 89th minute.
Palace struck back through Patrick van Aanholt to record their first-ever Premier League win at Old Trafford.
Although 21-year-old James felt his side should have gone on to claim the three points after his equaliser, he says they have the chance to bounce back next week when they take on Southampton.
"We want to redeem ourselves. This game is something that we’ll look back on and look at what we could’ve done better," he told the club's website."The Southampton game next week will be a completely different test and we’ve got to be ready for it."
He added: "After we scored, there were three or four minutes to go, the crowd were behind us and we were in control to go and get another.
"Unfortunately, they’ve hit us on the break and scored."
United had a chance to pull level earlier in the game when they were awarded a penalty, but Marcus Rashford saw his effort bounce off the woodwork.
And while James feels his team-mate was unfortunate, he believes his team had more chances to get back into the game.
“It was a great strike, I think he was unlucky,” he said.
“But, as you’ve seen in the game, we went on and got the equaliser and we were in control that whole second half. We should’ve come away with more.”
James' goal was his second for United since he joined from Swansea in the recent transfer window, having previously found the net in their opening game against Chelsea.
"Obviously, I’m delighted to get the goal and that’s something I want to keep improving on and keep doing," he said.
"But, at the end of the day, it’s a team game and we didn’t win the game today so obviously I’m disappointed."