The Hammers were 2-1 down at the London Stadium when they were awarded a late penalty.
Moyes chose to bring on club captain Noble to take the spot-kick due to his good record.
The 34-year-old had scored 38 of his previous 42 penalties, while not missing one since December 2016.
West Ham have struggled with spot-kicks of late having missed four of their last five.
Noble stepped up and United goalkeeper David De Gea, who himself had a terrible record for saving penalties, was able to palm to ball away.
"Do I regret it? Not at all," Moyes said after the game.
"That's what happens in management, you have to make big decisions. Sometimes they go for you, sometimes they don't.
"I just thought, we have a really good penalty taker on the bench. I'd have been more disappointed if I hadn't made the decision. If one of the other players had taken it and missed I'd have thought, 'Why didn't I bring Mark Noble on'?
"He has an incredibly good record on penalties. So with Mark going on, and De Gea not saving many, I'm hoping that is the equaliser."
The decision to send on Noble drew plenty of criticism, especially from Sky Sports pundits Graeme Souness and Roy Keane.
talkSPORT's Ally McCoist had no problem with the reasoning behind Moyes' call to bring on the midfielder.
He told Breakfast: "I understand totally why David Moyes is getting some stick for making the substitution, but I do not have a problem with it at all.
"We would be having the complete opposite conversation if he hadn't brought him on and he'd missed.
"It's just one of these things. I think I heard him on the radio after saying he'd only missed four penalties, therefore why wouldn't you do what he did.
"It didn't work out, so therefore you do leave yourself open for criticism. You've got to take it on the chin and move on."
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