Alvarez and Ishikawa, both left-handed hitters, took advantage of a stiff wind blowing out to right field at Hammond Stadium. Alvarez, the Pirates' third baseman, connected off former Pittsburgh teammate Kevin Correia.
"He's a buddy of mine, and I talked to him before the game," Correia said. "The wind was blowing out to right pretty good. I'm like, `You going to try to hit one?' He was like, `No way." And sure enough, he got it."
It was the second home run of the spring for Alvarez, who tied Arizona's Paul Goldschmidt for the National League lead last season with 36.
Ishikawa's homer came off Twins right-hander Vance Worley in the sixth. The two-run shot landed halfway up the batter's eye in center field.
Pirates: Right-hander Gerrit Cole allowed three hits and an unearned run in three innings of work. Cole, who hit 98 mph on the stadium radar multiple times, struck out four in his second official outing this spring.
"It was a step forward from last time," Cole said. "There were some good parts about it. There are still things that need to be cleaned up. I think it's important to keep throwing fastballs to build up arm strength and get your delivery in check."
Twins: Correia gave up four runs on seven hits. Correia, who also faced his former team in last Friday's "B" game, needed 31 pitches to escape a three-run third inning as he worked for the first time this spring with rookie catcher Josmil Pinto.
Pirates: Right-hander Jeff Locke remains day to day with right side discomfort. He threw two shutout innings in his only official spring appearance.
Twins: Center fielder Aaron Hicks (right elbow irritation) returned to action for the first time since March 4 and had a pinch-hit homer on the second pitch he saw.
Right fielder Oswaldo Arcia (stomach virus) returned to the lineup for the first time in eight days and struck out three times.
Shortstop Pedro Florimon took batting practice on the field for the first time since his Feb. 17 appendectomy and could appear in minor league games as soon as this weekend.
"We did our due diligence," Antony said. "We had some interest. We talked to his agent. I think what they were looking at and were looking for was different from what we were thinking."
Santana signed a one-year, $14.1 million deal with the Braves after they lost starter Kris Medlen to an elbow injury on Monday. The Twins were interested in a three-year deal for a reported $30 million to $33 million, but Santana preferred a one-year deal.
"We're trying to build something here," Antony said. "We weren't necessarily looking for a one-year guy. The Braves are in a different situation than we are. A one-year deal makes some sense for some clubs. I don't think it made sense for us."