Smith led off the seventh against A's lefty Daniel Coulombe, and the speedy outfielder dropped down a bunt that Coulombe fielded, then threw into right field. By the time the A's got the ball back to the infield, Smith stood on third. Corey Dickerson followed with a sacrifice fly to right to score Smith with what turned out to be the winning run.
Dickerson said his approach was to simply "get a pitch over the plate;" something he could "put up in the air."
"Just try to relax that at-bat, try and be more smooth and quiet and see the ball a lot better," he added. "And luckily, I saw the pitch right out of his hand, and I was able to put a good swing on it."
The Rays had to endure a high-wire act from Alex Colome, who allowed the A's to load the bases with one out in the ninth. But the Rays closer retired Jed Lowrie on a foul pop to third before striking out Khris Davis to end the threat and earn his 18th save of the season.
"Not at one point did we feel like it was unraveling," Rays manager Kevin Cash said. "The A's put together some good at-bats against [Colome]. Probably better at-bats than what we've seen a team do this year on back-to-back nights. It's going to happen. But the way he keeps his composure in those situations. And the 2-1 pitch he dotted down to Lowrie to get the count back to 2-2. Once he got the foul popup, all the pitches the he made to Khris Davis right there were just outstanding."
Stephen Vogt gave the A's an early lead with a two-run single in the second against Rays starter Chris Archer. Vogt doubled home another run in the fourth.
The Rays tied the score at 3 in the fifth. Smith, Dickerson, and Evan Longoria drove in runs against A's starter Jesse Hahn in the frame.
Archer allowed four runs on seven hits and three walks while striking out eight to earn a no-decision. Hahn also came away with a no-decision after allowing three runs on seven hits and two walks while striking out five in five innings.
"It's been frustrating," said A's manager Bob Melvin, whose club is now 9-23 in road games. "Granted, a lot of fight -- certainly in the ninth inning, as well. Some of the younger guys coming up with some key hits and some really good at-bats. But a few more at-bats, and two games could swing the other way."
MOMENTS THAT MATTERED Vogt delivers: Vogt, a former Rays catcher, has never been friends with Tropicana Field. He entered Sunday's action hitting .097 in his career at the Trop, the place that hosted some of his rookie struggles when he went 0-for-25 with the Rays. Sunday, Vogt got the A's going when he singled home two in the second inning against Archer to put the A's up 2-0. He added an RBI double in the fourth that put the A's up 3-0.
Double trouble: Down 3-0 with two outs in the fifth, the Rays got five consecutive hits against Hahn to score three runs. Of those hits, four were doubles. Daniel Robertson started it off with a double to left, then Smith doubled to center to score Robertson. Dickerson's double to left scored Smith before Longoria singled home Dickerson. Logan Morrison added a double to right for good measure before Steven Souza Jr. grounded out to end the inning with the score tied at 3.
QUOTABLE "Mallex Smith, I'd be happy if we don't need to see him for a while. He torched us." -- Vogt on Smith, who finished the four-game series batting .571 with a homer, three RBIs, four stolen bases and six runs scored
"Outstanding effort. It's been a long couple of days. Just really impressed about the way the guys continue to respond." -- Cash, after the Rays played four games in three days
SOUND SMART WITH YOUR FRIENDS • In taking the no-decision, Hahn extended his winless road stretch to 11 games. He has not won an away contest since June 2015 at San Diego, matching the longest skid for any Oakland pitcher in 17 years.
• Seven of Rasmus' home runs have come in the sixth inning or later. Sunday's home run brought his first game-tying homer since a ninth-inning, two-run homer on April 24, 2016, against Red Sox closer Craig Kimbrel.
WHAT'S NEXT Athletics: Following an off-day on Monday, Jharel Cotton will get the nod when Oakland opens a two-game series with the Marlins on Tuesday at 4:10 p.m. PT at Marlins Park. Though he has held opponents to a .169 average, Cotton has walked 11 in 17 1/3 innings. Cotton is 0-3 with a 6.04 ERA over his last four starts.
Rays: The Rays spend their off-day traveling to Toronto to face the Blue Jays on Tuesday at 7:07 p.m. ET. Jake Odorizzi will take the mound looking to keep the ball in the yard: He's given up at least one home run in all but one of his starts this season. Odorizzi gave up a three-run shot in an otherwise solid start against the White Sox on Thursday, a win in which he struck out eight over 6 2/3 innings.
Bill Chastain has covered the Rays for MLB.com since 2004.
Jeff Odom is a contributor to MLB.com based in St. Petersburg who covered the A's on Sunday.