Hannahan, who is competing for the starting third-base job with Lonnie Chisenhall, connected on a 1-2 offering from left-hander Tyler Skaggs, one of the D-backs' top prospects, to put the Indians up 7-2.
The Indians also used the long ball to grab an early lead off Trevor Cahill, who was making his Arizona debut after being acquired from the A's in the offseason.
Michael Brantley led off the game with a single up the middle and Jason Kipnis followed with a home run to the right of the batter's eye.
One batter later, Shin-Soo Choo hit a homer to put the Indians up 3-0.
"We came out swinging right off the bat," Indians manager Manny Acta said. "Kipnis and Choo set the tone early and we continued to swing the bat well throughout the day. The ball was jumping and we made good contact and made things happen. We also threw the ball well overall."
Cahill pitched two innings, allowing three runs on four hits, with three strikeouts.
"I never have done very well in my first couple Spring Training outings," Cahill said. "But it's just a matter of getting comfortable I think. My arm feels good so I think that's the biggest positive of the day."
The D-backs got on the board in the second when Paul Goldschmidt doubled to lead off the inning and scored one out later on Ryan Roberts' grounder.
Later in the frame, Gerardo Parra drew a two-out walk and came around to score on Henry Blanco's double to left to cut the Cleveland to 3-2.
That was the only damage that starter David Huff allowed. Huff, who is competing for the fifth spot in the rotation, tossed three innings.
Up next for Indians: After a rough spring debut, Indians starter Ubaldo Jimenez will take the mound for his second Cactus League outing when the Tribe hosts the Angels in a 3:05 p.m. ET tilt at Goodyear Ballpark. Starters Josh Tomlin and Jeanmar Gomez are also slated to appear in the game. Gomez is one of four pitchers vying for the fifth spot in the rotation.
Steve Gilbert is a reporter for MLB.com. Read his blog, Inside the D-backs, and follow him on Twitter @SteveGilbertMLB. This story was not subject to the approval of Major League Baseball or its clubs.