Daley Blind struck in stoppage time to cancel out Cheikhou Kouyate's brilliant 49th-minute volley.
Noble missed a glorious chance to hand West Ham a 2-0 advantage during the closing minutes and he felt his team-mates thoroughly merited a first win over Manchester United in 13 attempts.
The midfielder told Sky Sports: "It feels like a loss to be honest. It just shows how far we've come when we're gutted we drew with Man U at home.
"There were some world-class players out there today and I felt during many spells we were the better team. We just couldn't hold out until the end.
"There was a lot of heart, a lot of team spirit and we've got good players.
"Enner [Valencia] and [Diafra] Sakho up top gave everything for us, chased their centre-halves down and made it very difficult for them.
"It looked like three points towards the end but you can't switch off against these top teams because they punish you and that's what happened."
Louis van Gaal introduced Marouane Fellaini to the visitors' attack as they pushed for a point and Noble conceded that switch to a more direct approach caused problems for West Ham's defence.
"They brought on big Fellaini, got the ball on the halfway line and just stuck the ball on his head for the last 10 minutes, which was difficult," he added.
"Cheik [Kouyate] was struggling and Tomk [James Tomkins] was struggling as well. When it falls, it fell perfectly for Blind but he still had to finish it."
