Unfortunately, errors and a lack of run support kept the left-hander winless through his first three starts.
On Sunday it all came together for Rodriguez. Jordan Schafer hit his first career grand slam on a shot that bounced off the glove of Dodgers right fielder Andre Ethier, and the left-hander pitched three-hit ball through seven innings as the Astros romped past Los Angeles 12-0.
The Astros avoided a sweep and handed the Dodgers their most-lopsided shutout loss since a 13-0 defeat by the Angels in June 2004.
It's the largest shutout win for Houston since beating Colorado by the same score in July 2007, a game where Rodriguez also allowed three hits in seven innings.
Rodriguez (1-2) struck out six and walked three to lower his ERA to 1.42. Fernando Rodriguez and Brandon Lyon pitched an inning each to complete the shutout, which was the first of the year for Los Angeles.
The Astros committed four errors in each of Rodriguez's first two starts and were shut out in his last outing, a 1-0 loss to Washington.
"I feel good because last time I threw a good game but I didn't win," Rodriguez said. "Today I threw a good game and I won, so I feel very good."
Carlos Lee got things going for the Astros with two-run homer in the first inning. Schafer's slam came an inning later and was just the second career grand slam Chad Billingsley (2-1) has allowed in his seven-year career and the first since 2008.
"Wandy threw a great game today," Lee said. "We probably only needed one (run). He continues to pitch great and keep us in the games."
Rodriguez said the early lead made things much easier for him.
"I felt very comfortable when I saw the score at 6-0," he said. "I tried to make all my pitches, tried to throw on the corner and throw everything."
Billingsley, who entered the game 4-0 with a 1.78 and ERA in four career starts at Minute Maid Park, allowed four hits and nine runs -- five earned -- in 3 1/3 innings.
He has been dealing with a groin problem, but said that had nothing to do with his performance.
"It wasn't an issue," he said. "I just couldn't throw strikes. It's one of those days you just have to forget."
Matt Kemp, who leads the majors with nine homers and 22 RBIs, went 0 for 3 to break a 10-game hitting streak. It was only the second hitless game this season for Kemp, who is hitting .450.
Then came Schafer's grand slam that deflected off Ethier's glove and into the right-field stands.
"When he went up and came down I thought he had it, but luckily he didn't," Schafer said.
Ethier didn't think it was going all the way to the wall at first.
"It carried a little bit farther; I kind of under-jumped, too, because I didn't think it was going that far and I didn't want to over-jump on it," he said.
Schafer has reached safely in each of his 16 games this year, the longest string for the Astros since Craig Biggio started the 2000 season with a 17-game stretch.
A sloppy fourth inning by the Dodgers, who finished with a season-high three errors, helped the Astros add three more runs. Castro tripled on a long fly that sailed just past a sprinting Kemp, who lost his cap as he dashed up Tal's Hill in center field to retrieve it.
"Obviously our offense didn't do anything to take some pressure off the staff, and obviously we didn't do anything defensive-wise to come up with some big plays to get us out of the innings," Ethier said.
Reliever Jamey Wright plunked Johnson on the backside in the fifth inning and he later scored on Castro's single for a 10-0 lead.
Johnson and Altuve each had RBI singles in the eighth.
The Dodgers' 12-4 record is their best start since starting with the same record in 2005. ... Schafer's four RBIs on Sunday were a career high. ... Gordon turned 24 on Sunday. ... Uribe was in the starting lineup on Sunday after missing the last four games with an injured left wrist. ... The Dodgers open a series with Atlanta on Monday and the Astros travel to Milwaukee.