The unsettled Argentine, who coach Roberto Mancini said had told him he wanted to stay at the club, showed his importance with emphatic finishes either side of a Joleon Lescott header.
The win saw City leapfrog Arsenal and boost their chances of avoiding a playoff to reach Europe's elite competition.
The win moves Mancini's in-form side into a coveted automatic Champions League place with 68 points while Arsenal are fourth with 67 also from 37 games with one remaining.
Tevez, who is now joint top scorer in the Premier League with 20 goals, has been dogged by rumours about his future since he submitted a transfer request in December.
But Mancini told a post-match news conference Tevez had reassured the Italian he would still be at Eastlands to spearhead their European campaign next season.
"Every time we say he is important and a fantastic player," the City boss told reporters. "I think he will stay here next season. We don't have a problem. He told me that he wants to stay, he has a five-year contract."
TEVEZ DOUBLE
A pedestrian first half was brought to life after 14 minutes when Tevez fired City in front against the team they beat to lift the FA Cup at Wembley on Saturday.
The striker played a one-two with James Milner before beating twisting and turning past defenders Ryan Shawcross and Andy Wilkinson before firing into the top corner.
Lescott doubled City's advantage eight minutes into the second half when he rose above goalkeeper Thomas Sorensen and headed into an empty net.
Tevez then wrapped up all three points with a ferocious curling freekick from 30 metres.
City visit Bolton Wanderers on Sunday knowing a win will clinch third and a spot in the Champions League group stage while Arsenal, who suffered a surprise 2-1 loss at home to Aston Villa last Sunday, must travel across London to play Fulham.
"It's important now that we gather our strength for the game at Bolton," Mancini told the City website.
"We had a fantastic gift on Sunday from Arsenal and now everything is in our hands and we can't lose ths chance."
