Harrison was originally scheduled to pitch in Sunday's series finale against the Rays.
Right-hander Roy Oswalt will be making the start in Harrison's place in an attempt to give the 26-year-old more rest.
"We're just trying to give [Harrison] an extra couple of days," manager Ron Washington said. "He understands that last year we gave him a break and it helped him. He's the only one that's been on regular terms, so we're gonna do it right now and see where it takes us."
Harrison is coming off back-to-back losses to the Royals and Rays. He pitched a combined 10 innings and allowed 10 earned runs, along with two unearned, in the starts.
Oswalt, meanwhile, will be making his first start since Aug. 23, when he received a no-decision against the Twins. He most recently came out of the bullpen on Tuesday, allowing a run in 2 1/3 innings of work in a loss to the Royals.
The move, in turn, pushes the entire rotation back a day, giving even more rest to the other starters.
"Everyone benefits," Washington said. "It certainly wasn't something that was planned; we just decided to do it."
Pitching against Cleveland might turn out to be a blessing in disguise for Texas. The Indians feature a left-handed heavy lineup and conventional wisdom would suggest that Harrison, a southpaw, would fare better. It also sets him up to pitch against Seattle in the following series, a team he's done historically well against.
"That just happened to fall like that," Washington said.