"I was coming after him because he already had his homer tonight," Wickman said after yielding a run in the ninth before getting Mench to pop out with a runner on third for his fifth save.
Wickman thought he had Mench struck out on a 2-2 pitch.
"I wanted that punch-out," he said. "I was going for a strikeout. I figured if I went and got beat it would have been on good pitches. I made a lot of good ones to him."
Mench then fouled off three straight with a 3-2 count before popping to second baseman Ronnie Belliard -- as Wickman pumped his fist in exhaltation.
Wickman tied a career high with his 21st consecutive save since Aug. 6 and matched Doug Jones for the Cleveland club record with 129.
"There's huge satisfaction in that, but I've got to credit all the doctors and trainers in Cleveland," said Wickman, who missed all of 2003 and part of 2004 after having elbow surgery before recording a career-best 45 saves last season.
Mench was 0-for-3 before hitting an 0-2 pitch from Guillermo Mota just over the left-field wall to open the eighth after Cleveland had scored twice in the seventh.
He became the first right-handed batter ever to reach the plateau, joining Jim Thome, Don Mattingly, Dale Long and Ken Griffey Jr.
"When you say 'ever' in our game, that better get your attention," Texas manager Buck Showalter said.
Long, Mattingly and Griffey share the major league record with homers in eight straight games.
"I don't care about records, I care about winning," said Mench.
"We walked too many people," Showalter said after Texas issued seven bases on balls. "That was the difference. That's a tough lineup. They can do enough damage without us helping them."
"They didn't give in and showed great patience," said Indians manager Eric Wedge. "We battled back all night."
The Rangers closed to 6-5 on Mench's homer, but the Indians got the run back on an RBI double by Jason Michaels, who had three hits to snap a 6-for-38 (.158) slide.
"J-Mike came through in a huge way tonight," said Wedge.
The teams exchanged leads all night.
Hank Blalock had an RBI single in the first off Jason Johnson. Teixeira was out trying to score from second on the play. Plate umpire Brian O'Nora waited to see if catcher Martinez held the ball before signalling out -- triggering a protest from Rangers third-base coach Steve Smith, who was quickly ejected.
Texas got consecutive RBI singles from Phil Nevin and Blalock in the fifth for a 3-2 lead.
Cleveland tied it when Blake walked and scored from first on Michaels' one-out double.
Texas took a 4-3 lead on Gary Matthews Jr.'s sacrifice fly, but Cleveland answered with an RBI single by Aaron Boone.
Matthews' RBI came on a line shot to left-center with the bases loaded that center fielder Grady Sizemore caught with a dive.
"He probably saved two or three runs," said Teixeira. "If he doesn't make that play, I think the game is ours."
- Koronka, who had won his three previous starts, gave up six hits and four runs over 5 2/3 innings. The left-hander walked four and struck out two.
- Johnson allowed 12 hits and four runs over six innings, walking two and striking out one. The 12 hits tied a career high. He also gave up 12 at Tampa Bay on Aug. 9, 1999.
- Martinez has reached base safely in all 23 games this season and 38 straight since Sept. 17.