On Saturday, Oakland ended the Yankees' 13-game winning streak with a 3-2 victory - that also snapped the Athletics' six-game skid. New York has lost two in a row for the first time since July 22-23.
"There's no give-up in this team," Kemp said. "We're going to continue to fight to make this playoff push."
Kemp entered as a pinch-hitter in the seventh and finished 2 for 2 with his fifth homer of the season, with three of them coming against the Yankees.
Green (7-6) served up his 11th homer of the season.
"I'm not usually a guy that backspins balls to right field, and I especially know Oakland is tough to get it out, so the last thing I was trying to do was hit a home run right there," Kemp explained. "I saw a good pitch and put my best swing on it, and I think I was as surprised as everybody else was in the stands."
Deolis Guerra (2-1) pitched a scoreless eighth. Andrew Chafin recorded the final three outs to earn his second save of the season.
Oakland opened the scoring in the fourth on Mark Canha's RBI forceout. It was originally ruled an inning-ending double play, but the call was overturned upon replay review.
The run was unearned due to an error by third baseman Gio Urshela. He made two errors in the game.
New York tied it in the seventh with the help of some shaky A's defense.
With Anthony Rizzo at the plate and two outs, catcher Yan Gomes dropped a popup in foul territory that prolonged the inning. Rizzo then sent a routine grounder to third, but it went through Matt Chapman's legs, allowing Gary Sánchez to score from second and tie the game at 1.
Neither starting pitcher allowed an earned run. Yankees left-hander Jordan Montgomery tossed six innings, giving up just an unearned run on six hits with three strikeouts.
"Monty pitched great today," New York manager Aaron Boone said. "I thought he had a little bit of everything going. He was obviously very pitch-efficient. I thought it was a really strong six innings by him."
A's righty Paul Blackburn made just his third start of the year and pitched five scoreless innings.
"That's all we were looking for -- two times around (the lineup)," Oakland manager Bob Melvin said. "And it ended up being five innings, so that was fantastic."
JUDGE'S ON-BASE STREAK ENDS
Yankees OF Aaron Judge went 0 for 4, ending his 20-game on-base streak. He hit .384 with seven home runs and 21 RBI during that span.
Yankees: SS Gleyber Torres (sprained left thumb) began a rehab assignment with Double-A Somerset and homered in his first game, finishing 1 for 2 with two walks. Boone is hopeful Torres will come off the injured list Friday. "If everything goes well, that's kind of the plan," Boone said. ... RHP Domingo Germán played catch out to 120 feet this weekend and is getting close to a bullpen session. Germán has been on the IL since Aug. 1 with right shoulder inflammation.
Athletics: SS Elvis Andrus returned to the lineup after missing Saturday's game with a minor left arm injury. He finished 0 for 4.
Yankees: RHP Corey Kluber (4-3, 3.04 ERA) will come off the injured list to open a three-game series against the Los Angeles Angels on Monday night in Anaheim. Kluber has been out since May 25 with a strained right shoulder.
Athletics: LHP Cole Irvin (9-12, 3.68) is on track to start Tuesday's series opener in Detroit against the Tigers. Irvin (hip discomfort) exited his last start after just three innings but threw a bullpen session Saturday.
Kemp gives A's late lead with 2-run dinger
Tony Kemp smashes a pinch-hit, go-ahead two-run home run in the bottom of the eighth inning for the Athletics.