Harrison allowed four runs -- three earned -- on six hits over six innings in the Rangers' 9-8 loss to the Arizona Diamondbacks.
Russell Branyan hit a grand slam and drove in five runs for Arizona.
Harrison allowed four runs on four hits and four walks in four innings against Cleveland in his last start.
"I think I started out a lot better than the last one," Harrison said. "For the first five I got ahead and stayed ahead. It's been a while since I threw six innings but I should be able to do it."
Harrison is competing with Derek Holland, Tommy Hunter and possibly Neftali Feliz for the final three spots in the Texas rotation. Feliz and Hunter both pitch on Thursday in day-night split-squad games.
Much depends on Feliz, last year's American League Rookie of the Year as a closer, who has been stretched out to start but may yet end up in the Texas bullpen.
Harrison faced only three batters over the minimum in the first five innings before running into trouble in the sixth when Chris Young hit a two-run double, Xavier Nady had an RBI single and a fourth run scored on an error.
"We made that mistake behind him but he's got to be better than that in that inning," Rangers manager Ron Washington said. "He understands where he went wrong right there."
Still, Washington said outside of the final inning he saw everything he could ask for from a starter.
"That's the Harrison we've been seeing all spring," Washington said.
Diamondbacks starter Daniel Hudson, the team's No. 2 starter behind Ian Kennedy, gave up seven runs -- one earned -- on eight hits and two walks with three strikeouts in an abbreviated 2 2/3 inning outing.
Five of the six final hitters Hudson faced hit safely. The lone exception was Ian Kinsler, who was safe on a sacrifice attempt when first baseman Nady failed to reach around Kinsler to snare Hudson's throw to the first-base bag.
"I don't know if I was tipping pitches a little bit but everything I threw they just hit," Hudson said. "If I was [tipping pitches] we know it's not something major I need to fix. I just wish I could have stretched out a little bit more."
Arthur Rhodes, the Rangers' situational left-hander, gave up Branyan's slam in the seventh that completed Arizona's rally from a seven-run deficit. He also drove in the go-ahead run in the ninth off Alexi Ogando.
Branyan, who can opt out from his Diamondbacks contract if he's not assured a roster spot by Friday, has four homers and 16 RBIs in 18 games.
Diamondbacks SS Tony Abreu was a late scratch with tightness in his back. Rangers OF David Murphy started in center while Julio Borbon, the team's presumptive everyday center fielder, attended to personal business. Rangers RHP Brandon Webb scratched his fourth batting practice session Wednesday morning when he was unable to get loose before throwing. Texas is 1-7 against the NL West. The Diamondbacks have won five of seven after losing nine straight.