The moves came after defensive miscues from Kiner-Falefa cost the Yankees in their 6-5 loss Saturday to the Guardians.
Manager Aaron Boone said before Sunday's game he felt Kiner-Falefa was "pressing," which led to the decision.
"You know, playing a little bit not to make that mistake," Boone said. "And I think that's gotten in his way a little bit. So I just felt like I needed to do this today. I still expect him to play a huge role for us in what we hope is a couple more weeks of baseball, but I felt like today it was something that was the right thing to do."
Cabrera, an infielder by trade, was playing out of position in left field and failed to keep a ball in front of him in the ninth inning that turned a single into a double. He slotted back into his more natural position Sunday.
Boone said Cabrera would "probably" stay at shortstop for Game 5.
Hicks, meanwhile, saw his first action of the series. He went 0 for 2 with a walk.
Boone also reiterated before the game that his decision not to use Clay Holmes in Game 3 was due to the closer's recent shoulder issue and some minor soreness Holmes reported after his outing on Friday.
Game 5 will be played Monday in New York. The winner will face the Houston Astros in the American League Championship Series.
ESPN's Bradford Doolittle contributed to this report.