The New York Yankees selected Cave in the sixth round, with the 209th overall selection, of the 2011 MLB draft. He signed with the Yankees, receiving a $825,000 signing bonus. He made his professional debut that season for the Gulf Coast Yankees. In his first game, he fractured his kneecap in a collision at home plate. The injury caused him to miss the rest of the season as well as the entire 2012 season. In 2013, Cave played 113 games for the Charleston RiverDogs of the Class ASouth Atlantic League. He finished the year hitting.282/.347/.401 with two home runs and 31 RBIs. Cave started the 2014 season with the Tampa Yankees of the Class A-AdvancedFlorida State League. After hitting.304/.354/.395 with three home runs and 24 RBIs in 90 games with Tampa, he was promoted to the Trenton Thunder of the Class AAEastern League in July, where he finished the season batting.273 with four home runs and 18 RBIs in 42 games. Cave spent 2015 with both Trenton and the Scranton/Wilkes-Barre RailRiders of the Class AAAInternational League, compiling a combined.278 batting average with two home runs and 39 RBIs. He was selected by the Cincinnati Reds in the 2015 Rule 5 draft. Cave did not make the Reds' Opening Day roster, and he was returned to the Yankees on April 5, 2016. In 2016, he played for Trenton and Scranton/Wilkes-Barre, posting a combined.268 batting average with eight home runs and 55 RBIs, and in 2017, he returned to both the Thunder and RailRiders, batting a combined.305 with 20 home runs and 56 RBIs. The Yankees added Cave to their 40-man roster after the 2017 season but later designated for assignment on March 12, 2018 following the Yankees signing of Neil Walker.
The Minnesota Twins acquired Cave on March 16, 2018 for minor league pitcher Luis Gil. He began the season with the Rochester Red Wings of the International League, and made his MLB debut on May 19. He hit his first major leaguehome run in his first major league game. During the 2018 season, Cave hit 13 home runs and led all Major League Baseball hitters in average distance per home run, at 421 feet. On May 13, 2019, Cave was sent down to the minor leagues. He was added to the playoff roster for the 2019 postseason.
Personal life
Cave's father, Bryan, played baseball at Kecoughtan and became the head coach of The Apprentice School. Both of Cave's sisters played softball and were named All-District.