Tepesch (2-0) allowed two runs in 6 2-3 innings, striking out four to win consecutive starts for the first time in his career.
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 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.
Gimenez made it 6-2 with a two-run double down the line in left in the eighth off reliever Jared Burton.
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 surrendering runs in each of the first two innings.
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.
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).