In a dramatic finale at London Stadium, the home side were awarded a stoppage-time chance to get on level terms, after former Hammers loanee Jesse Lingard fired United into a 2-1 lead.
Noble is a spot-kick expert, having stuck away 27 of 31 previous attempts in the Premier League, but the 34-year-old had not taken a kick of the ball before being handed the responsibility of saving the game.
The substitute, sent on by Moyes after the penalty was awarded for Luke Shaw's handball, saw his shot to the right corner pushed away by David de Gea, who had conceded his last 21 penalties on league duty.
United held on for the closing seconds to see out victory, leaving Moyes with questions to face about the sense of his decision.
"I chose Noble to come off the bench and take the penalty kick because I know how good his penalty record is," Moyes said.
"Mark was brave enough, he wanted it, and so from that point of view there's no responsibility with him at all."
Moyes admitted West Ham lacked an obvious candidate to take the penalty, with captain Declan Rice having been poised to take responsibility before Noble came sprinting on.
"We just don't necessarily have someone who is a natural penalty kick taker. It is an easy thing to work on because you can stand and take penalty kicks in training ... but it's a different thing when you've got to go on the stage and do it," Moyes told a news conference.
The Hammers boss failed to complete a full season as manager of United in 2013-14, being sacked in April after struggling to live up to his billing after succeeding Alex Ferguson.
Since then, he has failed to earn a win in eight matches as a manager against the Red Devils, drawing twice and losing six games.
It was a cruel twist that it should be Lingard who scored the 89th-minute winner, and he appeared somewhat apologetic as the visitors celebrated.
Lingard revived his career during his spell at West Ham last season, scoring nine goals in 16 Premier League games. The England midfielder put a midweek Champions League blunder behind him on this occasion to give Ole Gunnar Solskjaer's side three points in the capital.
"Jesse got a second chance to bounce back as a player when he was at West Ham last year. We'll always be grateful for what he did for us," said Moyes. "I wish he hadn't scored, but we had a chance to get a draw out of it and we missed our chance."
Lingard's winner means West Ham have now had 47 players score both for and against them – excluding own goals – in the history of the Premier League, more than any other team.