The Gunners were up against it at half-time thanks to goals from Richard Dunne and Darren Bent at the Emirates Stadium.
But the same two players gave away penalties in the second half, both converted by Van Persie, sandwiching a fortuitous Theo Walcott strike, as the Gunners roared back for a 3-2 win.
"It was very important," Van Persie told ESPN. "At 2-0 down, it's always going to be a hard task to come back.
"But a crazy six minutes in the second half changed the game.
"Credit to the team. We all did it. Credit to all of them because we worked hard and we're still in it."
Van Persie said the team still believe they can win silverware this season.
"[It's] another chance, because we're still in the Champions League," he said. "We worked so hard. We are excited for the next round."
Walcott, whose goal came as a defensive clearance ricocheted off him into the net, said: "We need a bit of luck. Hopefully the luck's changing.
"The team worked so hard. This cup means so much to us. We want to win this trophy. We need to show the fans we're ready
"It shows we've got great characters in the dressing room."