Ben Zobrist drove in Alcides Escobar with the go-ahead run in the seventh inning as the Royals rallied from a three-run deficit to defeat the Astros 5-4 on Friday and even the American League Division Series at one win each.
Zobrist grounded a single to left on a curveball by Houston reliever Will Harris that scored Escobar, who led off the inning with a triple to right center.
The Royals play the next two games in Houston and on Sunday face left-hander Dallas Keuchel, who is 15-0 this season at Minute Maid Park.
"I just kept telling the boys we've got a long way to go, just keep plugging away," manager Ned Yost said after the Royals were trailing by three runs early. "Let it happen, don't try to make it happen. Just let it happen and it did."
Harris, the fourth of six Astros pitchers, took the loss.
The Astros jumped out to a 4-1 lead but managed just two singles after the third inning.
Kazmir departed after 5 1/3 innings, charged with three runs, five hits and a walk.
"From the first pitch of the game I felt I had good command of all my pitches," Kazmir said. "I was able to give the hitters a different look. I felt great. I felt I was getting stronger as the game was going on."
Oliver Perez replaced Kazmir after center fielder Cain's double. Perez faced three batters, retired none, and two runs scored to tie it at 4.
First baseman Eric Hosmer delivered a soft single to the opposite field to get Cain home. Perez yielded a single to designated hitter Kendrys Morales and walked Mike Moustakas on five pitches to load the bases.
Manager A.J. Hinch had seen enough of Perez, replacing him with Josh Fields, who walked catcher Salvador Perez on four pitches -- the last one nearly hitting him on the shoulder as he ducked -- to score Hosmer.
Cueto had a shaky first three innings, permitting four runs and six hits, including a home run by Rasmus, but only one hit after that. Cueto's final line was four runs, seven hits and three walks in six innings.
"I felt strong," Cueto said with coach Pedro Grifol acting as his interpreter. "I felt like I was aggressive. Obviously this was a very important game that I had to go out and perform for my team.
"I didn't feel any pressure at all. I knows the magnitude of the trade and I know why I was brought here. I take a lot of pride in my stuff and a lot of pride in helping my club win."
The Royals won another challenge in the ninth when Davis picked off pinch-runner Carlos Gomez at first base. After being called safe on the field, the call was reversed after a 62-second review.
The Astros again struck early and often, seizing a 4-2 lead after three innings.
Rasmus doubled with two outs in the first, scoring right fielder George Springer, who had walked. Rasmus has an extra-base hit in his first six postseason games, a major league record.
"I feel blessed," Rasmus said. "The Good Lord is with me. It's just one of those things. I'm just playing and not thinking about no records or none of that stuff. I'm just loving the game right now and trying to help these boys win a ball game."
The Astros got two runs in the second inning after Springer punched a two-run single to left with the bases loaded. That gave Springer eight RBIs in five games at Kauffman Stadium.
The Royals got a run back in the bottom of the second when catcher Salvador Perez homered to left.
Rasmus answered with a home run to lead off the third, driving a 93 mph fastball on a 1-2 count from Cueto out to right-center field. Rasmus has homered in three straight postseason games.
The Royals scored one run in the third when Zobrist grounded into a double play, which got right fielder Alex Rios home.
The Royals, however, scored three runs in the sixth and seventh inning to pull out the victory.
NOTES: OF Carlos Gomez was not in the Astros' lineup for the second straight game with a left intercostal strain. Jake Marisnick again started in center. ... DH Kendrys Morales became the third Royal with a multi-homer postseason game on Thursday. The others were George Brett (twice) and Willie Aikens. ... The Astros announced the roof will be closed at Minute Maid Park for games Sunday and Monday. ... Astros LHP Dallas Keuchel and Royals RHP Edinson Volquez are the Sunday probables.