Rios looks to help his new team maintain that lead Sunday, when the Rangers try for a seventh consecutive victory overall and eighth straight on the road over the Houston Astros on Sunday.
A day after Texas (67-50) acquired him from the Chicago White Sox, Rios went 2 for 3 with a game-tying triple in the eighth before scoring the go-ahead run in Saturday's 5-4 victory.
"I came here to try to contribute in any way possible and take advantage of everything that I'm given,'' Rios said.
The Rangers have won a season-high six in a row and 11 of 12 to take a one-game division lead over Oakland, which has dropped seven of nine.
Rios, who hit .277 with 12 homers and 55 RBIs for the offensively challenged White Sox, will attempt to fill the void left when slugger Nelson Cruz received a 50-game drug suspension earlier this week.
"The only thing I want him to make certain that he understands is that we didn't bring him here to carry us,'' manager Ron Washington said. "We brought him here to fit in. Whatever he can give us, we will take."
Washington has been able to lean on shortstop Elvis Andrus, who hit a two-run homer Saturday and is batting .337 in his last 22 games.
Texas has won seven straight and 17 of 20 at Minute Maid Park, where the Astros (37-78) have dropped four in a row and 15 of 18. The Rangers scored two runs each in the seventh and eighth innings, taking advantage of a Houston error in both.
"When you give a team extra outs, especially a team that has a good offensive team like the Texas Rangers, you put yourself at more risk to give up runs,'' Houston manager Bo Porter said.
After allowing 16 runs in 15 innings while going 0-2 during a three-start stretch, scheduled Texas starter Martin Perez (4-3, 3.81 ERA) has yielded three over 13 2-3 frames of the last two. He overcame a season-high four walks to give up two runs and strike out seven in 6 1-3 innings of a 5-2 win over the Los Angeles Angels on Monday.
"He's certainly been more under control and hasn't let the game get too quick," Washington told the Rangers' official website. "He's executing pitches, getting ahead early in the count and throwing his changeup for strikes."
This will be the left-hander's first appearance against the Astros, who counter with Dallas Keuchel (5-6, 4.96).
After his first two appearances this month came out of the bullpen, Keuchel returns to a rotation where he's gone 5-4 with a 4.50 ERA in 14 starts. The left-hander's last start came at Toronto on July 27, when he earned the win despite yielding six runs and three homers in 7 1-3 innings of an 8-6 victory.
Keuchel has a 2.81 ERA in three starts against the Rangers without factoring in any of those decisions.
Houston's Chris Carter was 5 for 54 in his previous 15 games before he went 2 for 3 with a homer Saturday. He is 3 for 4 with a home run and a double versus Perez.