Boston has won six games in a row and 11 of its last 13 as Rodriguez seeks to become the staff's second 11-game winner -- a day after Rick Porcello reached 11 wins with a victory that completed a sweep on Sunday at Kansas City. That gave the Red Sox a 7-2 road trip and a two-game lead over the New York Yankees atop the American League East.
Mike Minor goes Monday for the Rangers, who went 2-2 in Detroit at the start of a 10-game road trip. Texas closed its series in Detroit with a 3-0 win Sunday.
While the Rangers (40-51) are a mile away from contending, Shin-Soo Choo has become a story. On Sunday, he waited until there were two outs in the ninth inning to reach base for a 47th straight game -- breaking Julio Franco's club record.
The record hit, a roller to third base, also extended Choo's road streak to 18 games.
"I don't know how I did it -- 47 games already," Choo said. "A lot of people helped, especially my teammates."
Choo was named as a reserve to the AL All-Star team Sunday.
Boston saw J.D. Martinez and Mookie Betts win the fan vote while Craig Kimbrel and Chris Sale were named to the pitching staff and Mitch Moreland selected as a reserve. Andrew Benintendi is among the final vote candidates.
After two shaky outings (14 hits, nine earned runs in 10 innings), Rodriguez reset his special first half with six shutout innings (three hits) as part of the three-game interleague sweep at Washington last week. He is 10-3 with a 3.84 ERA and 1-0 with a 3.29 ERA in two career starts against the Rangers.
Rodriguez brought his changeup back into his mix in his last start -- and it paid off as the 11 a.m. game saw the Red Sox complete their sweep in Washington.
"A good mix today," Red Sox manager Alex Cora said after the game. "More changeups, a few sliders, great fastball. He competed the whole time under tough conditions. That's what makes him good.
"That changeup is one of the best in the league; he can throw it to righties and lefties. He was really good today."
Said Rodriguez: "The last few starts, I was throwing more fastballs and was getting more hard contact. Today, I was using my changeup and slider, and it worked well like before.
"Every time I throw the changeup for a strike and get a swing and miss, it helps my fastball, because they just think of the changeup and they get below it."
Minor comes in with a 6-4 record and a 4.63 ERA in 16 starts this season. He has gone 2-0 with a 2.53 ERA in his last five games.
He beat the Red Sox on June 3 in Texas and is 1-2 with a 3.86 ERA in seven games (three starts) in his career against Boston and 0-2 with a 4.50 ERA at Fenway Park.
The current Red Sox roster is 19-for-61 (.311) with three homers and 13 RBIs against Minor.
Martinez is 3-for-4 (.750) with a homer, newcomer Steve Pearce is 4-for-8 (.500) with a homer and Betts is 2-for-4, also with a home run. Jackie Bradley Jr. is just 1-for-8 (.125) against the left-hander.
The Texas roster is just 6-for-35 (.171) with 10 strikeouts against Rodriguez. Choo is 1-for-6 (.167).
The Red Sox placed Christian Vazquez (broken pinkie) and Brian Johnson (hip) on the 10-day disabled list and summoned two pitchers. William Cuevas (5-5, 3.65 ERA in 15 starts at Pawtucket) was recalled for the second time in a week and 30-year-old Ryan Brasier was selected in the midst of a 22-outing scoreless streak at Pawtucket. Austin Maddox was transferred to the 60-day DL to make room for Brasier on the roster.
With Johnson hurt after filling in for Steven Wright the last turn through the rotation, the Red Sox had not named a starter for Tuesday's second game of the series. Hector Velazquez (6-0) and Cuevas are said to be in the mix.
Martinez rested Sunday after fouling some pitches off a foot Saturday night but is expected to play Monday.
"He's a little banged up," Cora said of Martinez. "It made sense to stay away from him. We'll give him one day and he'll be back in the starting lineup tomorrow."