Their lead in the division over Boston was trimmed to 1½ games.
"It's huge. There's no question," Everett said. "You've seen it from the beginning of September to now. We've had games snatched from us from the ninth inning on. For us to snatch a game away from a team like Tampa Bay and their pitching staff, that's huge for us."
Longoria, who returned to the lineup last Saturday after being sidelined with a broken wrist, hit a two-run homer in Tampa Bay's five-run first against Glen Perkins. The All-Star third baseman added solo shots off Philip Humber in the fourth and Bobby Korecky in the seventh for an 8-6 lead.
Longoria joined Jonny Gomes as the only Rays to hit three home runs in a game in the club's 11-season history. Gomes also did it as a rookie, hitting three against Kansas City on July 30, 2005.
"Obviously I'm pretty happy about my performance, but not the outcome," Longoria said.
"I expect us to bounce back. We've been doing it all along. Win or loss, it's pretty much 15 minutes of celebration or 15 minutes of having your head down and figuring out what you can do tomorrow to help the team."
The Rays can clinch at least a wild-card playoff berth as early as Friday if they beat the Twins, and the White Sox, Blue Jays and Yankees all lose. They can also ensure their first postseason berth by winning two of the remaining three games in this series.
"I still like our position, absolutely," Wheeler said. "It's just a difficult one to swallow. That's all you can say."
The first-inning drive was Longoria's first homer since Aug. 5, two days before he fractured his wrist in Seattle. He has 25 on the year.
Dioner Navarro and Carlos Pena also went deep as Minnesota allowed five home runs in a game for the second time in five days. The Twins have given up a major league-high 33 home runs in September, 18 over the past five games.
"We can't seem to keep the ball in the ballpark, and we didn't do a good job of it tonight," Twins manager Ron Gardenhire said. "But our guys never quit. ... There at the end, Alexi put one in the seats and the rest of the guys did their thing."
Rays starter James Shields allowed six runs, five earned, and eight hits in 6 1/3 innings.
Jason Kubel hit a three-run homer and Joe Mauer went 3-for-4 with three RBI for the Twins. Mauer drove in the last two runs charged to Shields with a seventh-inning single off David Price that trimmed Tampa Bay's lead to 7-6 before Longoria went deep in the bottom half.
- Rays LF Carl Crawford, out since early August after undergoing surgery on his right middle finger, probably will not play again during the regular season. The club initially hoped he might be able to return for the final three games at Detroit.
- Rays CF B.J. Upton, who has missed the last nine games with a strained left quad, is improving and could return to lineup by the end of the weekend.
- Twins INF Brian Buscher jammed his left thumb during Wednesday night's loss at Cleveland and is day to day.