The Polish goalkeeper saved penalties from Gary Caldwell and Jack Collison to help the North Londoners book a place in next month's FA Cup final.
Everton leapfrogged Arsene Wenger's men into the fourth and final Champions League place in the table, but Fabianski hopes that the cup win can inspire his teammates to usurping the Toffees before the season finishes.
"It picks up the club and the fans, it gives us more belief that we can manage first of all to finish in the top four, and then hopefully to win the final," he told reporters.
"I think it's more for the fans, I think they have been waiting a long time and they show up every away game, win or lose they always follow.
"It was a great atmosphere today as well so it's great to do it for them."
Wenger's side host West Ham United at the Emirates Stadium on Tuesday night.