Ohtani was jammed by a 94 mph fastball from reliever Taylor Rogers, and the pitch hit his right ring finger as Ohtani struck out.
Simmons then tried to beat out a grounder on the right side of the infield. He landed awkwardly on his left ankle as he lunged for first base, and he had to be helped off the field.
X-rays afterward were negative for both players, but Simmons will undergo an MRI on Tuesday to determine if there was any ligament damage.
"More concerned now about Simmons' injury than Shohei's," Angels manager Brad Ausmus said. "I feel like Shohei by all indications should be fine, a quicker timeframe. We don't know what Simmons' timeframe is, but that's probably the bigger concern right now."
Simmons, a four-time Gold Glove winner at shortstop, bats cleanup and leads the team in hits (56) while batting .298 with 21 RBIs.
"I mean, it's tough, especially back-to-back like that," two-time American League MVP Mike Trout said of the injuries. "You guys saw the replay (of Simmons' injury). We've just got to hope for the best and hope it's not too serious.
"It kind of deflates you when you see guys go down, especially (Simmons), who has been having an unbelievable season so far. His reaction said it all. He obviously was in a lot of pain, so it's just tough."
Cahill, who is 1-4 with a 6.00 ERA in nine career games and seven starts against the Twins, has failed to finish five innings in five of his last six starts. He allowed six runs on six hits in 4 1/3 innings at Target Field last week.
Pineda, who is 1-1 with 6.08 ERA in four career starts against the Angels, comes in off an 11-6 victory at Seattle on Thursday night. He threw a season-high seven innings, allowing five hits, including three solo home runs, to snap a personal three-game losing streak. He has allowed 13 home runs in 47 innings this season, his first since undergoing Tommy John surgery in 2017.
The eighth inning on Monday night was much more enjoyable for the Twins, who have won six of their last seven games, thanks to a two-run homer by Miguel Sano that broke a 1-1 tie and turned out to be the game-winner.
"Just a good swing," Minnesota manager Rocco Baldelli said. "He stayed through the ball well and drove it out that way (391 feet to right-center). ... He has the kind of strength where, regardless where you're playing, he can drive the ball out anywhere. It's just a big hit. It's a game-changing type of at-bat for him and great for us."
The Twins improved to 16-8 on the road.
--Field Level Media