The Gunners faced their first season in 14 without Champions League football when Antonio Di Natale scored for the Italians in Udine but the North Londoners rallied to secure a 3-1 aggregate win and qualify for the group stage of the prestigious competition.
But van Persie wants that to be just the start for Arsenal who have lost two key players in quick succession this summer.
"Of course it's been a hard week, with Cesc [Fabregas] leaving, Samir [Nasri] leaving. We've been in the papers for the wrong reasons," he told The Mirror.
"But I think we played well at Udinese under immense pressure, so that's a positive sign. And now it's up to us to show that every three or four days, because that's football.
"It's not that we can live on this night for the rest of the month or the rest the year. We have to do it every single time.
"I think that we surprised a few people. Hopefully, we can do it more often."