"We've had a lot of close games like this that we couldn't quite pull off at the end," manager Mike Matheny said. "So, it doesn't matter to me. Just that we did."
Edward Mujica (1-1) worked a scoreless ninth for the Cardinals after Adams' pinch-hit, two-run homer tied it two innings earlier.
Jay drew a full-count walk off A.J. Ramos (3-3) with two outs in the ninth and took third easily on Robinson's pinch-hit single, then scored without a play after Stanton hesitated before throwing a relay that skipped under Logan Morrison's glove at first base.
"I was just trying to get the ball before it hit the ground," Morrison said. "I should have played it back or just let it go because it was on the line.
"We should have won that game, no doubt about it, but we didn't and now it's over and that's why we play every day."
Stanton did not speak to reporters after the game.
The Marlins got homers from Derek Dietrich and Morrison but their run of four straight series wins ended after dropping the first two against the Cardinals.
Manager Mike Redmond was ejected for arguing a close play at the plate in the fourth, with replays indicating Adeiny Hechavarria's legs crossed the plate before catcher Tony Cruz tagged him on the shoulder.
Redmond was already frustrated after an incorrect call at third base Friday ended up saddling the Marlins with an unusual double play in a 4-1 loss. He thought Hechavarria was "clearly safe" and wasn't certain that Cruz made the tag.
"I knew that run was going to be big," Redmond said. "You can only take so much, right? I think of those guys in the dugout and they're busting their butts. You've got to stick up for those guys, too."
Adams' homer off Mike Dunn foiled the Marlins' switch from starter to a lefty-lefty matchup and tied it at 4. Adams has both of the Cardinals' pinch homers this season and is 6 for 16 against lefties with two homers and six RBIs.
"In that situation, I don't know if I'd pull him for anyone," Matheny said. "He's earned it. If we're going to use him, we're going to use him."
Both starters reached season bests for innings, with Eovaldi going up three runs in 6 2/3 innings and Joe Kelly allowing four runs in six innings.
Matheny gave Kelly the fifth spot in the rotation on June 22 but the Cardinals didn't need him until now because of three off days, and the right-hander was used just once in long relief on June 28 before facing the Marlins. Matheny said Kelly will get another start next week.
Morrison has four homers in his last six games against the Cardinals. His fourth of this season put the Marlins up 3-1.
Kelly singled for his sixth career hit in 38 at-bats and scored on Matt Carpenter's triple in the third, a hooking drive that barely got past center fielder Marcell Ozuna. Carlos Beltran followed with an RBI single before Matt Holliday grounded into his 21st double play, by far the most in the majors.
Dietrich doubled with one out in the second and scored easily on Hechavarria's single.
The Cardinals are slotting ace Adam Wainwright ahead of rookie Shelby Miller on Tuesday, giving the 11-game winner two starts heading into the All-Star break but making him ineligible to pitch in the game. ... A second straight giveaway, this one for Holliday jerseys, attracted a second straight sellout with attendance of 45,475. Fans jammed entrances more than two hours before game time, to make sure they got the souvenir given to fans 16 and over. ... Cardinals SS Daniel Descalso had two throwing errors, a day after 2B Carpenter had a pair of errors. ... Cruz made just his eighth start of the season with Yadier Molina sent for medical tests on an injured knee. ... Cardinals starting SS Pete Kozma, in an 0 for 17 slump, did not play for the third straight game.