The Gunners started the night behind Tottenham Hotspur but knowing that maximum points in their final two games would secure UEFA Champions League qualification.
There was nothing to choose between the sides at half-time as Lukas Podolski's opener was cancelled out by Shaun Maloney, but Walcott restored Arsenal's lead at the start of a three-goal blitz in eight minutes which sealed the win and also confirmed Wigan's relegation.
"It was one of my most important goals and hopefully there can be a few more next weekend," Walcott told Sky Sports.
"I am so proud of the guys because we deserved it.
"I felt like we had to take the initiative and make runs in behind, which we weren't really doing in the first half.
"The final ball wasn't the best at times and we weren't really putting people in the box, but in the second half we did that and it showed in the 4-1."
Arsenal head to Newcastle for their final match of the season and Walcott is delighted that they are still in control of their own destiny.
He added: "It is in our hands, we always knew that was the case."