The England forward started up front and missed a number of chances before netting the opening goal to reward manager Arsene Wenger's faith.
"I was pleased with that goal," Walcott said.
"Luckily I managed to grab one out of my many chances
I was disappointed because I should have got many, many more, but I'm getting the chances so I'm happy with that.
"When you get given the opportunity you have to try to grab the goals when you can.
"We create chances so we just have to put them away
We're starting to get back to the old Arsenal."
It was arguably Walcott's most difficult chance that he scored after he missed the target from close range twice previously.
Francis Coquelin won the ball in midfield, fed Mesut Ozil to play a delightful pass over the top and Walcott brushed off Marc Muniesa before firing beneath Jack Butland.
Walcott recognised the contribution of Coquelin in the goal.
He added: "It's things that people might not notice, like the tackle from Francis.
"He's in that position for a reason, you can see it there, and that just started off the rhythm of the movement.
"Mesut's ball was fantastic, and it was all down to the touch
It put the defender out of the game."
Wenger suggested Walcott and Olivier Giroud, who came on to score the second, will be rotated in the coming days, with Dinamo Zagreb and Chelsea the opposition this week.
Walcott added: "I've said to the manager that I'd like to play up front.
"I can play in any of those three positions up front, so it's a good option for the manager to have.
"It's nice to see Olivier among the goals as well, which shows it's great competition that we have for places here."
Butland's performance was one of two positives for Stoke boss Mark Hughes
The other was the return from long-term injury of Bojan Krkic.
Stoke have just two points after five games, but Butland cannot put his finger on any significant problems - although he says Stoke could be more aggressive.
The goalkeeper said: "There's lots of things
There's no major issues
It's just a case of looking at ourselves and making sure when we come into a game everyone's ready.
"We've done the work, it just comes down to make sure we implement it on the pitch and ultimately starting winning 50-50s, start being more aggressive and being better when we perform."
Chances were at a premium for Stoke as Butland repelled the Gunners time and again.
He added: "For almost 80 minutes it was still 1-0
Whether we were good enough or not we were still in the game.
"We didn't create enough clear-cut chances, Arsenal certainly had more.
"We've got to look at ourselves and do enough and we didn't warrant getting anything from the game in the end."
Source : PA