Arsene Wenger’s side were forced to dig deep for the three points following Carl Jenkinson’s red card, and Walcott was delighted they showed their battling qualities.
“It was an important win,” the England international told arsenal.com. “The three points have us on a high and we’ve got a nice week now without any midweek game.
“A lot of players are back from international duty and need a bit of a break. Some may be rested for the FA Cup game, but that’s just as important as we want to go far in that competition. If we lost [at Sunderland], we would have been dwelling on it for so long, so I’m glad we didn’t.
“We’re very confident [of finishing in the top four]. We can battle through any sort of situation. The teams around us are winning as well, but we’ve just got to make sure we look after ourselves and try to win as often as we can.
“We’ll see where we are at the end of the season but we’ve got to show the belief we did at Sunderland. I felt there were a lot of winners in that dressing room and the boys were fantastic.”