The forward struck twice in Everton’s thrilling 3-3 draw with Pep Guardiola’s side, though a stoppage-time Jeremy Doku goal denied them victory.
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Moyes’ men had roared back with three goals in 15 minutes, and Barry made a major impact off the bench, replacing Beto in the 64th minute to spearhead the attack.
“It’s always a privilege to play against the big teams, like Man City,” he said after the game.
“I’m very happy to come on and score two goals.
“We drew, so we need to keep pushing. We have just three games now so we need to take more points.”