''You have to bring your 'A' game because he's really tough,'' Jimenez said. ''If you make a lot of mistakes you're probably not going to win.''
Jimenez (3-2) walked four in the first three innings but retired 13 of the final 15 hitters he faced. The right-hander struck out six and walked five.
The Indians did enough on offense to hand Darvish his first major league loss. Asdrubal Cabrera hit a two-run double and Jason Kipnis homered off the Japanese star, who struck out a career-high 11 over six innings.
Darvish (4-1) signed a $56 million, six-year contract with Texas in January. The Rangers also bid nearly $52 million to gain negotiating rights to the 25-year-old right-hander, who was 93-38 with a 1.99 ERA in Japan.
Chris Perez pitched the ninth for his 11th consecutive save. He gave up one hit and was helped by a diving catch in right field by Shin-Soo Choo on a sinking liner by pinch-hitter Adrian Beltre.
The Indians took two of three from the two-time defending AL champions after going 1-9 against the Rangers in 2011.
''It's no secret they whupped our butts last year,'' Perez said. ''It was good to kind of prove to them we're going to be here and stick around.''
Cleveland has won three straight series and leads the AL Central.
Jimenez, who had allowed 14 earned runs in 21 2-3 innings over his last four starts, acknowledged he was frustrated with his recent performances.
''You don't feel good when you're not pitching well, but I knew I couldn't keep my head down,'' he said. ''I knew I had to keep working hard every day.''
Cleveland manager Manny Acta was relieved by Jimenez's outing.
''He pitched with some confidence and some flair,'' Acta said. ''It was good to see him do that.''
And the fact that it came against a team with such offensive firepower made the outing more impressive.
''It's not like he was facing the lowest batting average in the league,'' Acta said. ''He was facing the leading hitting club in the American League right now.''
Jimenez was helped out of a two-on, one-out jam in the first inning when catcher Carlos Santana threw out Josh Hamilton trying to advance to second on a pitch in the dirt. Jimenez then struck out Michael Young with a runner on third.
In the third, Jimenez issued three straight two-out walks, but got out of it when Young grounded out.
Jimenez used a good breaking ball to put away Texas hitters. He retired 10 in a row between a leadoff double by David Murphy in the fourth and a one-out walk to Mitch Moreland in the seventh. Jimenez finished off his day by striking out Yorvit Torrealba and Alberto Gonzalez.
Texas scored twice in the eighth off reliever Tony Sipp to get within 4-2.
Darvish had a little bad luck and fell behind 3-0 in the third.
Johnny Damon reached on a single when his infield popup fell safely as second baseman Ian Kinsler lost it in the sun.
''It went up in the air, into the sun, and dropped,'' an upset Kinsler said before walking away from reporters.
Acta's mood was much sunnier after Cleveland's second straight Sunday win aided by glare. One week ago, nine-time Gold Glove outfielder Torii Hunter lost a fly ball hit by Cabrera in a win over the Los Angeles Angels.
''We got another sun ball,'' Acta said. ''It was a Sunday sun ball back-to-back.''
Darvish walked Kipnis, then Cabrera grounded a double inches inside the first-base bag, scoring both runners.
Darvish struck out Travis Hafner and Santana before Choo hit a grounder to the left side. Rangers shortstop Elvis Andrus made a nice, backhand stop in the hole and a leaping throw to first, but sailed the ball into the stands. Choo got credit for a single, and the error scored Cabrera to make it 3-0.
Kipnis hit his fifth homer leading off the fifth, extending his hitting streak to 10 games. It was only the second homer allowed by Darvish.
Darvish gave up three earned runs and six hits, walking four.
''He's a big kid with a very good arm,'' Acta said. ''What impresses me the most is his secondary stuff. I don't know how many pitches he has and I don't care, but guys take some awful swings against him. We have a lot of guys in the league who can throw in the mid-90s (mph), but not many who can spin the ball like that.''
NOTES: Andrus went 1 for 2 with two walks. He has hit safely in all 26 career games against Cleveland. ... Cabrera is hitting .385 (25 for 65) in his last 16 games. ... Beltre's three-run homer in the 11th inning Saturday was the first allowed by a Cleveland pitcher to a pinch-hitter in extra innings since Hall of Famer Brooks Robinson connected it for Baltimore off LHP Dave LaRoche on April 19, 1977. It was the first pinch-hit homer given up in extra innings by an Indians pitcher at home since RHP Frank Funk served one up to Kansas City's Leo Posada on June 12, 1961.