Walters was called up from Triple-A Columbus on Sunday when outfielder David Murphy and designated hitter Nick Swisher went on the disabled list with injuries. The home run was the fourth of his career.
Cody Allen (4-2) worked around a two-out walk in the ninth.
Trevor Bauer, pitching against the team that took him with the third pick of the 2011 draft, allowed two runs and struck out nine in a career-high eight innings.
Bauer didn't allow a hit until the sixth when Arizona scored to break a scoreless tie. Michael Brantley's two-run single put Cleveland ahead in the bottom of the inning, but the Diamondbacks tied the game in the seventh on Aaron Hill's RBI groundout.
Bauer faced the minimum 15 batters through five innings before running into problems in the sixth. Jake Lamb drew a leadoff walk before Xavier Paul lined a clean single to right, putting runners on first and third. Lamb scored when Jordan Pacheco hit into a double play.
Bauer clashed with both Arizona management and teammates, leading to a short tenure with the Diamondbacks. The right-hander was 1-2 with a 6.06 ERA in four starts for Arizona in 2012 and was traded to the Indians in December.
Nuno allowed two runs in 5 2/3 innings.
Tuesday night's game was postponed in the top of the fourth after a 3 hour, 40 minute rain delay, forcing the team to play a doubleheader. The Diamondbacks announced Tuesday that left-hander Andrew Chafin would make his major league debut in the first game, but manager Kirk Gibson opted to go with Nuno.
Chafin is the scheduled starter in Game 2 against Cleveland's Josh Tomlin.
Diamondbacks: OF A.J. Pollock (broken right hand) is expected to resume his rehabilitation assignment at Triple-A Reno over the weekend. Pollock returned to Phoenix after being hit by a pitch on the same hand during his rehab debut in the Class A Arizona League, but tests showed no additional damage.
Indians: DH Nick Swisher (sore right knee) will be examined by Dr. Neal S. ElAttrache, the team physician for the Dodgers, on Tuesday in Los Angeles. A determination on whether Swisher needs surgery could be made at that time.
Diamondbacks: After the doubleheader, Arizona continues its 10-game road trip with a four-game series in Miami on Thursday. Arizona RHP Chase Anderson (7-4) takes on RHP Brad Penny (1-0). Anderson is unbeaten in his past five starts, posting a 2-0 record with a 1.74 ERA over 31 innings.
Indians: Cleveland is off Thursday, but continues its five-game homestand Friday against Baltimore. Cleveland RHP Corey Kluber (13-6) faces LHP Wei-Yin Chen (12-4). Kluber is tied for the American League lead in wins and ranks third with a 2.46 ERA and 187 strikeouts.