"I left a pitch up, and he did what good hitters do," Stults said of the Kipnis home run.
McAllister (1-0) allowed five hits with no walks and seven strikeouts.
"He was good. He came out of the bullpen with a fastball-changeup combination that was really strong," Indians manager Terry Francona said. "And he pretty much used it the entire way. That's probably one of the better changeups we've seen him have, and he established it early in the game and then he really threw his fastball and located it. He threw it up when he wanted to, down."
After working five scoreless innings, Padres starter Stults (0-2) ran into trouble in the sixth.
Stults allowed two runs (one earned) on five hits in 5 2/3 innings. He walked two and struck out one and got nine groundouts.
"He pitched a really good game. He can add and subtract. We had a chance in the second inning and didn't cash in," Francona said. "We had a chance to add on, and we didn't. Some of that, you tip your cap to the pitcher, because he did a really good job."
It was a far cry from Stults' first start of the season against the Marlins on April 4, when he allowed five runs on seven hits in four innings.
"For the most part, today was a better outing than the first one in Miami," Stults said.
The Padres went into this series wanting to get two starts from lefties against the Indians' lefty-dominated lineup. Robbie Erlin, another lefty started the second game.
"It's a team where a left-hander can maybe neutralize the left-handed hitters some. The game plan was to go after them and get some ground balls," Stults said of the Indians.
Ryan Rayburn had two hits for the Tribe (5-3).
The Padres put two runners on with two outs in the eighth inning but reliever Cody Allen struck out Seth Smith looking to end the inning.
Nick Hundley, Amarista and Smith had doubles for the Padres (2-6). Amarista had two of the Padres six hits.