Before that, the Indians right-hander only ran into trouble in the fourth when he was touched for back-to-back, one-out doubles by Anthony Rizzo and Alfonso Soriano, Rizzo scoring the first run of the game. Despite the consecutive hits, Jimenez pitched out of the inning by getting a pair of groundouts.
"I felt really good," Jimenez said. "I was able to be around the zone, attack the strike zone and get ahead most of the time. I feel like I'm ready for the season. I have a lot of confidence out there about throwing strikes. It doesn't matter if I fall behind in the count. I trust myself to come back in the count."
Jimenez is ticketed to start the second game of the season on April 3 at Toronto. He should have one more Cactus League start on Friday against the Royals at Surprise Stadium. Jimenez came into the game with a 2-0 record and 4.92 ERA to show for his five spring appearances, four of them starts.
He allowed his only walk and a single to open the eighth before being removed from the game. In all, he gave up three runs on four hits, and was in the strike zone all game, walking the one and whiffing three.
Francona had told Jimenez that he would face two batters in the eighth, and then go to the bullpen and finish his day. Jimenez threw 100 pitches, including his bullpen.
"He was good," Francona said of Jimenez. "The progression continued. He could have stayed in. He continues to build. I thought it was a tremendous outing. He got out to the eighth inning. He's building endurance. He pounded the strike zone. He continues to work on his delivery and the results speak for themselves."
Meanwhile, Cubs left-hander Chris Rusin, already ticketed for Triple-A, kept the Indians offense at bay and allowed only singles to Mike Aviles and Nick Swisher when he left after recording the first out of the sixth inning. He walked three and whiffed three, but the Indians never moved a runner against past first base against him.
Rusin was being considered for the Cubs bullpen but was told he's more valuable as a starter. He'll be in Triple-A Iowa's rotation this year.
"I'll go down and work on all the stuff I need to work on and be ready for when they call on me," Rusin said.
It helped him that his was his second big league Spring Training camp. When he walked into the clubhouse, he felt comfortable from Day 1.
"This year, I obviously had a couple more starts than last year," he said. "Getting my feet wet last year was a big help. If this was my first Spring Training, I probably wouldn't be this way. Since I had experience last year, I knew what to expect. I've got to keep throwing strikes."
Up next: Brett Myers is slated to get the start against the Rockies in a road game on Monday at 10:10 p.m. ET at Salt River Fields, which will be available on MLB.TV. Roster moves are expected to be made shortly, with a decision on Jason Giambi before a Tuesday deadline when he can become a free agent. Asdrubal Cabrera missed Sunday's game after tweaking his back in the cages. He's not expected to be back in the lineup on Monday, Indians manager Terry Francona said.