The 27-year-old Morton, a veteran of 75 big-league appearances, was a winner for the ninth time in 2011 on Aug. 8, when he twirled eight innings of six-hit shutout ball to defeat San Francisco, 5-0.
It extended an already career-best win total for the New Jersey native, a third-round pick of the Atlanta Braves in 2002.
However, Morton is winless in three tries at No. 10, including a 4 1/3-inning effort at St. Louis on Aug. 25, in which he was tagged for nine hits and five runs in an 8-4 loss.
He is 2-4 in seven career starts against Houston, the most recent being an 8-2 loss in Pittsburgh on July 6.
Rodriguez (10-9) scattered four hits, allowed three runs and walked four batters to get into the double-digit win column for a third consecutive season.
Jose Altuve tallied three hits and an RBI, as Houston won its second straight following a 1-6 stretch.
Ross Ohlendorf pitched the first five frames for the Pirates in the opener of this three-game series, but Jason Grilli (1-1) couldn't hold the advantage. Jose Veras was also touched for a pair of runs in the decisive seventh inning. Andrew McCutchen hit a two-run homer for the Pirates, who have dropped seven of their last 10 contests.
Houston sends 26-year-old righty Henry Sosa to the mound for his fifth career start.
The Dominican made his major-league debut on Aug. 10 at Arizona and dropped a 6-3 decision after allowing four runs on six hits in six innings.
Subsequent home starts against the Chicago Cubs and San Francisco yielded a loss and a no-decision, before Sosa got win No. 1 with a 3-1 defeat of the Giants in California on Aug. 25.
He allowed a run on four hits in six innings, bringing his combined totals through four starts to 23 innings pitched and 13 runs allowed on 21 hits. He's walked nine batters and struck out 16.
The Pirates have still won nine of 13 versus the Astros this year.