"We didn't start that well," Van Persie told Setanta Sports. "Newcastle were sharper, better, they wanted it more.
"It was tough. Especially when we scored we thought we could take it easy a little bit but that wasn't the case because Martins scored within a minute so after that we knew we had to crack on and at least score one more goal."
The win completes a good week for Arsenal who came back from a goal down against Hull in midweek to progress to the FA Cup semi-finals and Van Persie believes they can use those results as a springboard.
He added: "It tells us that we can bounce back. We showed it against Hull last week, we showed it again today.
"I think this team has a lot of character and it's up to us to show it until the end of the season because we can win lots of things this year."