Walcott's first-half brace ended up being the difference at the Emirates Stadium in what was a mature display from a side who fell to Olympiakos after a stunning Premier League win over Leicester City at the same stage last season.
"You could just tell tonight that the intensity was there from the start. It died off towards the end but that's obviously understandable. But then again, we kept the clean sheet - that's what we want to do.
"The first half again was fantastic. We knew after the Chelsea game that it couldn't just stop there and everyone's got to be proud because it's not easy to play game after game every three days.
"We just always want to start off the game with so much pace. As soon as we do that it puts a team on the back foot and you don't want to be playing against Arsenal when we're firing on all cylinders, and we did tonight.
"We know how well offensively we are, but everyone is doing their defensive job and everyone's working hard for each other. That's the experience we've got in this team now, and it's paying off."