Pence ended a slump with a home run and Berkman also homered to help the Astros beat the Arizona Diamondbacks 8-3 on Sunday.
"If you're going to be a good ball club you can't rely on one or two guys to carry the team," Berkman said. "This is the most balance lineup we've had since I got here."
Bourn scored Houston's final run when he hit a run-scoring single, stole second and third and scored after waiting for third baseman Mark Reynolds to throw to first on a soft grounder by Tejada.
"He pretty much stole home the way he waited," Berkman said. "That was the textbook definition of manufacturing a run."
Brian Moehler (3-4) scattered six hits over six innings for the Astros, who won for the fourth time in five games. The right-hander allowed three runs, struck out two and walked two for his third win in four starts.
"It was bend but don't break," Moehler said. "We won the series which is a good start to the road trip."
The Astros have won five consecutive series for the first time this season.
"I've said it and I'll keep saying it," said Astros manager Cecil Cooper. "Good baseball is the point. Good baseball is what we've got to play and what we've been playing."
"It's good we're playing hard and working our tails off to turn this around but the wins haven't come as well as they should have," Hinch said. "We have to find a way to get wins. It's mentally grueling. But I've said it before and I'm sorry for saying the same thing over again. The rest of the league doesn't care. It doesn't feel sorry for us."
The Astros wasted little time against Diamondbacks starter Billy Buckner (2-3).
Pence was hitless in his last 12 at-bats before homering.
Bourn walked to lead off the game and scored when Miguel Tejada's looping liner into right field skipped under Justin Upton's glove and rolled to the wall for a double. Buckner struck out Pence before Berkman homered onto the porch overhanging center field to make it 3-0.
"I got us down early and we were never able to come back from that," Buckner said.
Drew's two-out, two-run homer off Moehler in the bottom of the first cut the lead to 3-2. Pence hit a two-out home run in the third to extend Houston's lead to 4-2.
Reynolds led off the fourth with his 17th home run to pull the Diamondbacks within 4-3.
Buckner allowed six runs and five hits over five-plus innings with three walks and five strikeouts.
Michaels started in place of Carlos Lee, who left Saturday's game with cramping in his left leg and was given the extra day off. ... Rodriguez started for the Astros and tied Bob Boone for second all-time in games played as a catcher with 2,225. Rodriguez needs two games behind the plate to pass Carlton Fisk for the all-time mark. ... Bourn stole his team-leading 21st and 22nd bases before scoring in the seventh. ... Buckner's double was his first career hit. ... Tejada added his major-league leading 23rd double leading off the sixth.