"That's the kind of result we're looking for," Astros manager Cecil Cooper said.
Backe (5-8) went 6 2/3 innings, allowing two runs and six hits with five strikeouts and four walks. Wesley Wright replaced Backe with two on and two out in the seventh and struck out Carl Crawford to end the inning.
"It's nice to have a win under my belt. I know it's been a long time," said Backe, who won for the first time since May 23.
Hunter Pence also homered for the Astros, who took two of three from the surprising Rays after losing seven consecutive series. The Astros, who are 5-18 in their last 23 games, started the series mired in an eight-game losing streak.
"We got good hits at the right time," Lee said. "To take two of three from this team that's been so good at home, it's unbelievable."
The Rays got a two-run homer from Evan Longoria, but came up short in their bid to win 10 straight series at home. Tampa Bay, 30-13 at home, was coming off a three-game sweep of the NL Central-leading Chicago Cubs.
"We had opportunities," Rays manager Joe Maddon said. "You're not going to get it done all the time. I'm satisfied with the preparation and intent."
All three games in the series were decided by one run.
Kazmir (6-3) gave up three runs and four hits in 5 2/3 innings. He struck seven and walked two during a 106-pitch outing. The left-hander had allowed just two runs over 30 2/3 innings in five previous home starts this season.
Longoria put the Rays ahead 2-0 with his drive in the second. The rookie third baseman has 12 homers in 65 games since being called up from Triple-A Durham on April 12.
Pence then made it 2-1 when he opened the fifth with a solo homer. The shot ended Kazmir's season-long streak of not allowing a homer at home at 34 2/3 innings.
"They're a good hitting team," Navarro said. "They can hit the ball out of the ballpark and that's what they did today."
Houston second baseman Kazuo Matsui left in the sixth with a strained right hamstring. He was hurt after hitting a grounder to short.
"I just had a little pain [after running to first]," Matsui said through a translator. "It's felt tight lately. I'll have to wait and see how it feels tomorrow."
Astros RHP Shawn Chacon was dropped from the starting rotation and was sent to the bullpen. ... Lee had two homers and five RBIs in the series. He has 46 RBIs in his last 49 games. ... Tampa Bay 1B Carlos Pena (broken finger) is scheduled to take batting practice Tuesday and could return within a week. ... Maddon said RHP Edwin Jackson dropped his appeal and started serving a five-game suspension for his part in Tampa Bay's bench-clearing brawl at Boston on June 5. ... LHP David Price, taken first overall by the Rays in the 2007 amateur draft, is joining Double-A Montgomery. Price went 4-0 with a 1.82 ERA in six starts at Class A Vero Beach ... Rays CF Rocco Baldelli (chronic muscle fatigue) will continue to DH and is adding pinch hitting as part of his rehab program this week for Vero Beach. His status will be re-evaluated next weekend. ... Tampa Bay optioned OF Justin Ruggiano to Triple-A Durham.