Ross allowed just four hits, striking out eight and walking three.
The Rangers did not commit to Ross after the game but his performance left little doubt.
''He was pitching out there again today,'' Texas manager Ron Washington said. ''He still out there. He's done nothing to change that. He's been pitching his butt off. He's been doing everything. We have to make a decision soon. Robbie's done his job against a pretty good lineup. You can't ask for much more.''
Michael Choice hit his fourth home run, connecting off Indians starter Josh Tomlin.
Tomlin is competing with Carlos Carrasco for the fourth spot in the rotation. Danny Salazar will start the fifth game.
Tomlin allowed two runs in the first but put together four scoreless innings until Choice hit the home run.
''I made a couple mistakes and that first inning got away from me. After that I was able to settle back in,'' Tomlin said. ''I threw a hanging curveball. It kind of backed up on me.''
Indians manager, Terry Francona, also faces a tough decision. Carrasco pitched well on Monday. The home run didn't faze Francona.
''Just like the rest of them as they got stretched out, fatigue set in,'' Francona said. ''That's part of the program. I think Tomlin is as hard working as anyone here. That will never change.''
Nick Swisher had a double and a single against Ross. Swisher was scheduled to get extra at-bats as the designated hitter on Monday but miscommunications about use of the DH between the Reds and Indians sent Swisher home early.
Rangers: Ross is pitching for a spot in the Ranger's rotation. This was his fifth game and third start. Coming into the game Ross has allowed three earned runs in 14 1/3 innings for a 1.88 ERA. He added the seven scoreless innings against the Indians.
''He had velocity, hit his spots and changed speeds,'' Washington said. ''It's the best I've seen him pitch.''
Indians: Josh Tomlin, like Ross, is competing with Carlos Carrasco to start the fourth game of the season for the Indians. Tomlin missed most of last season with elbow reconstructive surgery. Tomlin has a 2.57 ERA in his prior four spring appearances.
Tomlin pitched 6 1/3 innings and gave up four runs on seven hits and two walks. He struck out five.
''I pitch every game like my job's on the line,'' Tomlin said. ''This is as good as I've felt since 2011.
Rangers: Catcher Giovanni Soto will miss three months with a torn lateral meniscus in his right knee. He will have surgery on Wednesday. ... Outfielder Engel Beltre has a fracture of the tibia on his right leg. He will open the season on the disabled list. ... Shortstop Elvis Andrus was the designated hitter against the Indians. He is on a throwing program caused by recurring soreness in his right forearm. He was examined on Friday. ... Yu Darvish has a stiff neck.
Indians: Jason Giambi will be placed on the disabled list on Sunday. He suffered a fractured rib on his right side after being hit by a pitch thrown by Edwin Jackson of the Cubs on March 7. ... Catcher Matt Treanor will start the season on the disabled list with a strained left hamstring.
Darvish was scratched from his opening day start for Texas. Darvish complained of a stiff neck causing Ron Washington to look for another option to start the season. The Rangers made the decision to scratch Darvish and send him back to Dallas to be examined. He will likely start the season on the disabled list.
SANTANA WILL PLAY THIRD
Lonnie Chisenhall was competing with Carlos Santana at thirdbase. The spring-long experiment to move Santana to third was a success. The Indians told Chisenhall that he made the team on Sunday. Santana will play some thirdbase this season.
''Carlos will play third,'' Francona said. ''I'm not sure how it will play itself out (as far as playing time) but this gives us a lot of options. Carlos has shown that he can play third and catch a major league game at the same time. That's a phenomenal skill set.''
Texas added right-handed pitcher Daniel McCutchen to the major league camp. McCutchen appeared in three games this spring. The Rangers signed catcher Chris Snyder to a minor league contract. Snyder was released by the Washington Nationals on Monday.