Smyly experienced discomfort in his left index finger while pitching to Rob Refsnyder leading off the fifth. The lefty was pulled and will have an MRI on Sunday.
"It happened on a pitch," manager Gabe Kapler said.
Caleb Baragar (2-0) later recorded the final out in the inning.
Rangers second baseman Rougned Odor left the game in the bottom of the third with right oblique tightness.
Solano's single made it 4-2 in the third.
Mike Yastrzemski drew a bases-loaded walk -- one of his four free passes -- in the sixth to give the Giants insurance and Solano singled in another run in the eighth.
Jordan Lyles (0-1) walked five and allowed four runs on three hits in four innings making his first start of the year for Texas.
The Rangers will try to avoid a sweep Sunday after losing 9-2 on Friday, which started a six-game stop in the Bay Area. Texas is set to play in Oakland next week.
"It wasn't our best game. It's frustrating when you have 11 walks and a hit by pitch," Rangers manager Chris Woodward said.
The Giants are 14-3 against the Rangers in San Francisco, and also beat them in both home game during the 2010 World Series.
With Smyly's untimely exit, San Francisco has now gone its initial nine games without a starting pitcher completing five innings, the longest such streak to start a season in major league history, according to STATS.
Brothers Scott and Tyler Heineman walked out to home plate from opposite dugouts and represented the Rangers and Giants to confirm the lineup cards, then posed for a photo with the umpires.
Scott homered in the Texas ninth and his brother watched him round the bases. Tyler doubled in the eighth.
Rangers: OF-INF Danny Santana missed his third straight game with tightness in his forearm near the elbow, a concerning issue that will have the Rangers be extra cautious and he is unlikely to throw for about a week. He was available as a left-handed hitter off the bench and to pinch run but can't play defense. The Rangers have discussed placing him on the injured list. "It definitely has been brought up but I don't think we're ready to make that decision just yet," Woodward said. ... "He says he feels pretty good but we're still going to be careful with it."
Giants: OF prospect Hunter Bishop, who tested positive for the coronavirus before camp resumed, is close to ready to return. He posted on Twitter: "I'm completely healthy and strong! Obviously the protocols and testing are what is limiting from stepping foot on the field -- but I'm all about safety and the health of the players present. Hopefully back ASAP." Kapler added no specifics other than: "Hunter said it all. The ASAP that he put out there is really well said."
Giants C Rob Brantly cleared waivers and was outrighted to the club's San Jose alternate site.
LHP Kolby Allard (0-0) makes his first 2020 appearance pitching the series finale Sunday for the Rangers while the Giants didn't announce a starter.
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