Just over an hour after a news conference to announce his eight-year contract extension, Miguel Cabrera singled as part of three consecutive Tigers hits to open the game. Rajai Davis hit a leadoff single, took second base on a wild pitch and scored on a Kinsler single.
Moore regrouped after Cabrera's liner into shallow center to strike out Victor Martinez and Torii Hunter, but he gave up a second run when Austin Jackson's ground ball through the left side sent Kinsler home from second. That was the theme of the day for Moore, who alternated nasty pitches with damage control on his way to three runs on seven hits over three innings.
It was nevertheless encouraging for Moore, who didn't walk a batter after issuing 10 free passes over his previous two starts.
"For me, personally, it's a little bit easier to walk away from a game like this, where I was in a zone," Moore said. "They beat me. I didn't put them on. Try to work around some of those base hits, and we'll be ready for the season."
Kinsler's second run off Moore began with a leadoff single before two wild pitches put him in position to score on a Martinez liner into left. Moore escaped further damage when Brandon Guyer stretched for a diving catch in left-center field, robbing Nick Castellanos of an RBI double.
Avila's home run came off Joel Peralta. It was just his fourth extra-base hit in a Spring Training that saw him bat .306 (15-for-49) with four RBIs. Dominic Ficociello added an RBI single in the seventh.
Up next: Right-hander Chris Archer will start Saturday afternoon, when the Rays travel to Montgomery to play the Biscuits, the organization's Double-A affiliate, in a 3:05 p.m. ET contest. For Archer, it will be his final spring start before making his first start of the season Thursday night against the Blue Jays at Tropicana Field.
Jason Beck is a reporter for MLB.com. Read Beck's Blog and follow him on Twitter @beckjason. This story was not subject to the approval of Major League Baseball or its clubs.