Puig, following a tie-breaking single by Alex Castellanos in the seventh inning, crushed a line-drive homer off former Dodgers farmhand Fernando Nieves.
Cleveland starter Cory Kluber, competing for a spot in the rotation, matched Beckett with three scoreless innings, scattering four hits and striking out two.
"I felt good from the get-go today," Kluber said. "I don't think I really have the luxury to try to work on stuff. I think I have to use each outing to try to show something, so to speak, because I am competing for a spot.
"I try to step back and not really think about it," Kluber continued. "The way I look at it is if I go out there and do my job and do what I need to do, that stuff will kind of take care of itself. It's not something I can control other than going out and performing and pitching well."
"He was good," Indians manager Terry Francona said of Kluber. "He was attacking the zone. He had some swing and misses with his fastball. He was aggressive. Good. Good tempo."
Francona was also impressed with bullpen candidate Cody Allen, who, after his outing today, has pitched three shutout innings and allowed just one hit with four strikeouts this spring.
"Stay out of his way," Francona said. "He's what you want. He's quick to the plate, he pounds the strike zone, and he's got great stuff. You just pat him on the back and say, 'Way to go, kid.'"
The Dodgers broke the scoreless tie with a two-out rally in the fourth inning off Vinnie Pestano. Luis Cruz, in his final game before joining Mexico's team for the World Baseball Classic, singled, took second on a wild pitch and was singled home by A.J. Ellis. Cruz and Ellis had two hits each.
Cleveland tied the game in the seventh with an unearned run off Matt Magill. Gomes reached first on an error by Justin Sellers, was singled to third by Mike McDade and scored on Juan Diaz's sacrifice fly.
With Nieves on the mound, Dodgers non-roster catcher Wilkin Castillo bunted for a single leading off the seventh inning. With one out, he advanced to second on Nieve's wild pickoff and was singled home by Castellanos.
Matt Kemp, in his second game back from shoulder surgery, went-0-for-3 with a strikeout and a flyout to the warning track.
Up next for Indians: Cleveland's rotation battle will continue on Monday, when the Indians hit the road for a 3:05 p.m. ET tilt against the Cubs in Mesa, Ariz. Right-handers Carlos Carrasco, Zach McAllister and Trevor Bauer -- all vying for a spot at the back end of the Tribe's starting staff -- are each scheduled to log three innings. Righty Alberto Cabrera will get the nod for Chicago. A live webcast of the Cactus League contest will be available on Indians.com.