Sanchez struck twice in the final 17 minutes against the Black Cats as the Gunners overcame a frustrating evening in front of a sparsely-populated Emirates Stadium.
Arsenal still need either Manchester City or Liverpool to falter on the final day if they are to qualify for next season's Champions League but thanks to Sanchez they have at least taken the battle to the very last.
Thousands of empty seats greeted the players as the game got under way, many staying away as part of a boycott organised in protest against Wenger signing a new deal and remaining in charge beyond the FA Cup final meeting with Chelsea.
Sanchez finally broke the resistance of the already-relegated Black Cats, first turning home a Mesut Ozil cross from close range and then heading in after Jordan Pickford had once again saved from Olivier Giroud.
Sanchez was only passed fit to play hours before kick-off and, having reached 23 goals for the campaign, Wenger was full of praise for the man whose own future remains uncertain, with his contract at Arsenal expiring in the summer of 2018.
"With him, I feel he had something still left (injury-wise), but once he's on the pitch and gets the ball, he always becomes a devil and forgets his pain," Wenger said of Sanchez.
"Nobody questions his desire and his quality.
"It shows as well that if you look at his numbers, he has developed here as a player and has become a top-class player and has not wasted his time here
Hopefully that will last for a long time."
Wenger has never finished outside of the top four in a full season at helm since his appointment in 1996.
The fact he holds such an achievement in high regard has been used against him in the past as Arsenal failed to win a title since their 2004 unbeaten campaign.
And the 67-year-old confessed surprise at the amount of importance others are now placing on his side achieving the same this year to qualify for the Champions League.
"I answered for 20 years the question, 'the top four is nothing special'
So I don't know why suddenly it could become such a big problem," he said.
"I'm quite surprised
I want to absolutely make sure that we are in there but after that let's get to 75 points and see what happens."
Wenger had suggested earlier in the week that some teams are already "on holiday" as the season draws to a close.
But Sunderland boss David Moyes claimed those remarks were ill-judged.
"I think that's an insult to football as I've been a player myself," he said
"I would hate it if anyone would question me if I was going to chuck a game away
We didn't play well on Saturday, we know that but I think you can see Sunderland had a go, tried to score at times but we were outclassed by a better football team."
Source : PA