Tepesch (2-0) gave up both his runs in the first two innings before lasting 6 2-3. He struck out four to win consecutive starts for the first time in his career.
''That's one of the things that made me successful today, to be able to put the first two innings behind me,'' Tepesch said. ''Treat each inning as a new inning.''
Alex Rios added two more hits and now has six straight multi-hit games. The Rangers have won three straight and five of six after a rough start to the season.
''Early in the season everything didn't go the way we wanted it to,'' Andrus said. ''At the same time, we still trust in each other and we knew there's a lot of baseball left. Everyone is feeling well right now as a team.''
Trevor Plouffe homered and Eduardo Escobar had two hits and an RBI for the Twins, who have lost four in a row since climbing two games over .500 on May 21.
After Kevin Correia (2-6) set down nine straight, Roughned Odor and Shin-Soo Choo lined back-to-back two-out singles in the fifth. Andrus followed with a double to the gap in left-center that scored both runners and broke a 2-all tie.
''It's two-out hits. That's been the theme this season,'' Correia said. ''That one pitch or two pitches a game, you make that one pitch and it's a world of difference.''
Gimenez made it 6-2 with a two-run double down the line in left in the eighth off reliever Jared Burton and Texas was on its way to another win.
''We just got off to a bad start,'' Texas manager Ron Washington said. ''I'm so happy that we're finally putting it together as a group.''
Minnesota had its opening day outfield back together for the first time since the first week of the season. Josh Willingham and Oswaldo Arcia were activated from the disabled list after Sunday's game and started alongside center-fielder Aaron Hicks.
But the re-enforcements weren't enough to overcome Tepesch, who didn't allow a runner past second base after the second inning.
Twins manager Ron Gardenhire knows it might take a few more at-bats before Willingham and Arcia are able to help lift Minnesota's struggling offense.
''That can be a huge part of it, driving the baseball and let them get a few at-bats here,'' Gardenhire said. ''It takes a little time, a little different pitching up here than down there.''
Tepesch dominated Triple-A Round Rock after getting demoted in spring training and has a 2.95 ERA in three big-league starts since being recalled May 14.
''He's shown these three starts what he's capable of doing,'' Washington said. ''But you can't hang your hat on a couple of outings. If he's around here healthy, he's going to get many outings. I just want him to be consistent.''
Hicks, a career switch-hitter, informed manager Ron Gardenhire before the game that he has decided to only hit from the right side. The struggling prospect is hitting .250 right-handed compared to .149 from the left side.
He had two singles Monday.
Two replay reviews went against the Twins. Arcia slided into third after a botched play in the outfield in the first, but a review showed third-baseman Adrian Beltre tagging him before his foot reached the base.
Gardenhire challenged an out call at first on Willingham's dribbler down the line in the sixth, but the call stood.
NOTES: Hicks told Gardenhire before the game he had no confidence swinging from the left side. ''He came to me this morning and said he wants to hit right-handed. So, obviously, when a guy has no confidence left-handed, you gotta do what you gotta do,'' Gardenhire said. ... This was Ron Washington's 1,186th game as Rangers manager, tied with Bobby Valentine for most in franchise history. ... Texas P Tanner Scheppers, out since April 18 with elbow inflammation, had his first rehab outing on Saturday with Double-A Frisco and will make three more with Triple-A Round Rock before being eligible to return on June 3. ... Choo hopes to return to left field on Tuesday after DHing the last two games with a sore left ankle. ... Phil Hughes (5-1, 3.15 ERA) will try for his fifth consecutive win on Tuesday against the Rangers' Yu Darvish (4-2, 2.35).