However, the Gunners may regret a host of chances missed which could have made the second leg in Rome a more comfortable proposition.
Van Persie was fouled by Philippe Mexes and then converted the penalty.
"I think it's a good result - it was a tough game," van Persie told ITV4.
"We had chances especially in first half, and in the second half as well we had some - but we're glad with a 1-0 win."
Gunners boss Arsene Wenger admitted he would have liked more goals to take to Rome.
"We played well," he added on ITV4. "There was room for more goals, that's the only regret from tonight.
"We kept a clean sheet, that's important, and I felt every time we got the ball we used it well."
He added: "It is enough, it's down to us to make it enough. You look at all the results at that level and they're close.
"We're the only ones who won tonight. It's better than a draw."