The Chilean's hat-trick inspired the Gunners to a 5-2 win at Leicester on Saturday.
Walcott netted Arsenal's first as they came from behind at the King Power Stadium and are now three points behind Barclays Premier League leaders Manchester United, who they host next Sunday.
Sanchez had not scored for the Gunners since their FA Cup final win over Aston Villa in May and Walcott now expects him to produce the goods.
"I am sure he is going to flourish now with many, many more important goals
It has been a long time coming, he knows that himself
But he is very pleased," he said.
"He is a top-quality player and even when he is not scoring goals he works so hard for us
He tracks back and does a lot for the team."
Walcott impressed in a central forward role and has now scored 12 goals in his last 13 Arsenal starts and believes it was one of his best displays since a serious knee injury last year.
"It will be up there since the knee injury and, before that, the Tottenham game," he told the club's official site
"I have always said I want to play up front
The manager has had faith in me and I just want to repay him."
Olivier Giroud wrapped up the win while Leicester's Jamie Vardy grabbed a brace, and hit the woodwork twice, to make it six goals for the season, one more than he scored in the whole of last term.
The former Fleetwood striker admitted he is now in the form of his life but wants results to back it up.
"It is, it's in the Premier League and I've never been here before except for last season but it's disappointing we didn't get the result," he told Foxes Player.
"I am pleased but it means nothing
Hopefully I'll be able to get more goals next week and we can take the points.
"There's disappointment in the dressing room and we need to get back onto the training field and focus on Norwich."
Source : PA