A point was the least Kevin Keen's side deserved from the Academy League fixture - rearranged after it was postponed a few weeks ago.
Hammers made the livelier start and took the lead thanks to a superb strike from Noble after 25 minutes.
The young midfielder played two reverse passes with team-mates before producing a brilliant chip from the edge of the penalty area that left the Palace keeper clutching at thin air.
Following wonder-strikes from Danny Bunce and Hogan Ephrain in recent weeks, it appears that Kevin Keen's team are holding their very own Goal of the Season competition!
However, Noble's strike was cancelled out just five minutes later, when a hopeful punt forward caused confusion in the Hammers' defence and a Palace striker nipped in to convert from an acute angle.
The hosts then took the lead 10 minutes after the break, but Hammers fought back and enjoyed the majority of possession before finally getting their reward 10 minutes from the end, when Noble pounced on a loose ball in the penalty area and drove it home.
The visitors then pushed forward for what would have been a deserved winner, but Palace held on for a share of the points.
Said Kevin Keen afterwards: "We played very well and a draw was the least we deserved for our efforts.
"We are a little bit disappointed by the result, because I felt the game was there to be won, but I was pleased with the performance - it is probably the best we have played since the winter break.
"There were some very good performances - Mark Noble played well on top of scoring two goals, while San Lee is also back to his best.
"I have to give Palace some credit, though, they are a strong side and made it very difficult for us, so all in all it wasn't a bad day."
Team: Behcet, Wright, Parrington, Blewitt (Henry 80), Bunce, Collington (Welsh 75), Tucker, Noble, Cohen, Lee, Cole (Yao 45). Subs unused: Reed, Singh.