The London club looked set to have to settle for a point at the Emirates Stadium on Wednesday as they struggled to break down a spirited and resolute Southampton side.
Fraser Forster produced sensational saves to deny Danny Welbeck twice and Olivier Giroud as Arsenal desperately pushed to force a winner which would move them up to sixth place.
And it duly arrived one minute from the end, when Chile forward Sanchez was on hand to ram home his 14th goal of an impressive debut season in England.
The victory leaves Arsenal two points adrift of fourth-placed Manchester United and Wenger praised the impact his big-money signing has made since his move from Barcelona.
He told Arsenal's official website: "It's difficult to find examples of someone who has settled so quickly. He's had a fabulous impact on the team."
Wenger acknowledged Arsenal were not at their fluent best, but the Frenchman still felt his side deserved a third successive win.
He said: "Today was the third game in seven days and we had to dig deep and be patient against a very good, well organised team.
"There was something missing in our fluency but it's important that we can win when it's not perfect.
"We gave very few chances away and we created more, so we deserved the win."
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