Rodriguez went 2-for-3 and drove in two runs. His solo home run, his 31st, broke a 2-2 tie in the sixth inning.
"It seems like every time I pitch well enough to win, he does something," Meche said.
Rodriguez hit grand slams in Meche's previous two victories against the White Sox and the Orioles.
Rodriguez's homer off Mark Portugal helped the Mariners salvage the finale of a three-game series. Rodriguez went 6-for-12 against the Red Sox.
"He had a great series, didn't he? He's a player," manager Lou Piniella said.
Boston lost for the second time in nine games.
Meche (4-2) allowed two runs and three hits in seven innings. He retired his last eight batters.
Rodriguez chased Portugal (7-10) with a two-out, RBI single in the seventh. The hit scored Troy O'Leary.
"Portugal was still pitching well when he came out. Seattle has a tough offense. I thought his command and control got better as the game went on," Red Sox manager Jimy Williams said.
The Mariners made it 2-all in the fourth. Rodriguez reached on shortstop Garciaparra's error and stole second before scoring on Ryan Jackson's single. Jackson was recalled from Triple-A Tacoma before the game.
The Red Sox took a 2-1 lead on consecutive doubles in the third by Brian Daubach and Garciaparra. Daubach went 1-for-3 with an RBI, and finished a torrid series in which he was 9-for-13 with two home runs and 12 RBIs.
Daubach had reached base seven consecutive times before Ibanez made a sliding catch in left field in the fifth.
©1999 The Associated Press. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed