Gardner and Happ become eligible for free agency. Gardner will get a $2.5 million buyout rather than a $10 million salary. Happ's deal did not have a buyout.
Britton, a 32-year-old left-hander, was 1-2 with a 1.89 ERA in 20 appearances this season, getting eight saves and filling the closer role when Aroldis Chapman was sidelined by COVID-19 from the start of the shortened season until Aug. 17.
His deal calls for salaries of $13 million next year and $14 million in 2022. In addition to the team's two-year option, Britton's contract included a $13 million player option for 2021.
Gardner, 37, has been with the Yankees since 2008 and is their last player from the 2009 World Series champions. He hit .223 with five homers and 15 RBIs in the shortened season, down from .251 with career highs of 28 homers and 74 RBIs in 2019. The Yankees agreed to a deal with a $2 million signing bonus and an $8 million salary for 2020, which became $2,962,963 in prorated pay.
Gardner lost playing time in 2020 to Clint Frazier, who hit .267 with eight homers and 26 RBIs.
Happ, a left-hander who turned 38 on Oct. 19, was acquired from Toronto in July 2018, went 7-0 in 11 starts and was rewarded with a two-year, $34 million contract. He went 12-8 in 2019 and struggled at the start of this season, prompting the Yankees to skip his turn.
He got stronger as the season went on and finished with a 2-2 record and 3.47 ERA in nine starts. His $17 million option originally would have become guaranteed with 27 starts or 165 innings, but the threshold was reduced to 10 starts with the shortened season and he fell one short.
Happ entered in the second inning of Game 2 of the American League Division Series against Tampa Bay and gave up two-run homers to Mike Zunino in the second inning and Manuel Margot in the third of the 7-5 loss to the Rays, after which he said he would have been more comfortable starting.
The Associated Press contributed to this report.