The Red Sox scored twce in the ninth inning to tie it up before giving up the game-winning in the bottom half of the frame. Matt Barnes suffered the loss in relief as he allowed one earned run on two hits with two walks without recording an out.
"I'm thinking, 'I've got to finish this game, '" Correa said.
The Red Sox won the series finale against the Astros for a second consecutive weekend, with Houston claiming the season series 4-2.
The Red Sox are 7-1 in the last eight Rodriguez Sunday starts.
After Roberto Osuna (3-0) loaded the bases, Christian Vazquez hit a two-run single for Boston to tie it at three.
"It's not his arm", Cora said. I know where we're at, I know where they're at.
The Astros answered with Gurriel's RBI single in the bottom of the seventh. It's about execution and there were certain days we didn't execute it. I went out there and tried to take the ball and (Cora) didn't like what he saw, so he took me out of the game. Rodriguez (5-3) was instrumental in keeping the Houston lineup shackled, allowing four hits while issuing only one walk against five strikeouts.
"(It's) not as dramatic as we feared", Hinch told reporters Saturday, adding that while Springer will likely miss a significant amount of time, the team feels it "avoided a catastrophe".
Houston recalled outfielder Derek Fisher from Triple-A Round Rock to replace him on the roster. Price has a career 2.06 ERA in 15 starts against Cleveland.
The Red Sox open a three-game series against the Astros starting at 8:10 p.m. ET on Friday.