Sanchez (1-1) allowed nine hits in 6 2/3 innings and struck out only one, but Houston stranded eight runners and didn't score until the seventh. Chris Leroux pitched a scoreless eighth and Leo Nunez struck out the side in the ninth.
The 26-year-old Sanchez allowed at least one single in every inning, but did not walk a batter. He threw 65 of 89 pitches for strikes, letting the Astros put the ball in play.
"I just wanted to attack the zone," Sanchez said. "I just tried to keep the score close. No one hit it hard, so I was just happy to throw strikes."
Florida manager Fredi Gonzalez said he's been preaching to his pitchers the importance of avoiding walks.
"Usually, the big innings are two hits strung in a row, with a base on balls, and another hit," Gonzalez said. "It's hard to string three or four hits in a row. It happens, but it's difficult. Those big innings, there's always a base on balls mixed in, or two."
Chris Coghlan walked and Hanley Ramirez singled to start Florida's early rally. Jorge Cantu singled to extend his hitting streak to 20 games, dating to last season. Lee charged in, but the ball slipped under his glove, allowing Coghlan and Ramirez to score.
Cantu is hitting .316 (25 for 79) during his hitting streak, which dates to Sept. 29.
"It's a great accomplishment," he said. "I can say I'm a little bit surprised and it's obviously unexpected. But I'm just going to continue to ride this wave and see how long it takes me."
Cantu went to second on the play, advanced on a wild pitch and scored on Dan Uggla's groundout to make it 3-0. Paulino escaped more trouble when shortstop Tommy Manzella made a spectacular diving catch on John Baker's blooper to end the inning.
"I got a real good read of the play off the bat," said Manzella, a rookie highly regarded for his defense.
Paulino settled down after the 29-pitch inning and struck out the side in the fifth. He reached 100 pitches in the sixth, though, and Florida added to its lead.
With one out, Cantu reached on a sharp grounder that Blum couldn't handle. Uggla followed with a double to left, prompting a mound visit by Houston pitching coach Brad Arnsberg.
Baker walked to load the bases for Ross, who drove in two with a single up the middle.
"We have to take advantage of every single error out there," Cantu said. "If they give it to us, we've got to take it."
"I had a bad inning, the first inning," Paulino said. "But after that, I made the adjustments I needed to make. Mentally, I felt great, and I did the best I could in the game.
"Nothing is going my way right now," he added, "but hopefully sometime, we will start to see results."
Humberto Quintero led off the bottom of the seventh with a single and Sanchez hit Manzella with a pitch, finally showing signs of fatigue. Pinch-hitter Cory Sullivan grounded into a fielder's choice before Michael Bourn singled to drive in Houston's only run.
Cantu has a 16-game hitting streak this year, the longest to open a season since 2008, when Nate McLouth hit safely in his first 19 games. ... The Astros wore green caps to commemorate Earth Day. ... Astros reliever Alberto Arias is out for the season after right shoulder surgery on Thursday. Arias, 26, appeared in three spring training games before he was placed on the disabled list March 26. General manager Ed Wade said the surgery repaired Arias' rotator cuff and labrum and removed calcium deposits. ... Ross has 10 RBIs in 16 games.