Tway played college golf at Oklahoma State. In his freshman year, he won the NCAA Central Regional and was named honorable mention All-American, first-team All-Big 12 and was selected to the All-Central Region squad.[4]
Tway turned professional in 2011 after graduating from Oklahoma State. He made his professional debut at the 2011 Travelers Championship, with his father caddying for him.[5] He played on the Web.com Tour in 2013 and picked up his first win in July at the Albertsons Boise Open, defeating Spencer Levin on the first playoff hole with a birdie.[6] Both had shot a tournament record 261 (–23). Tway finished 5th on the regular season money list to earn his PGA Tour card for 2014.[7]
During the 2014 season, Tway made only eight cuts in 23 events with no top-10 finishes. His best finish was a tie for 26th at the 2014 Travelers Championship, and he finished the season 177th in FedEx Cup points, losing his PGA Tour card.[8] Back on the Web.com Tour in 2015, Tway carded two top-10s, with a runner-up finish at the Mexico Championship,[9] but he finished 28th on the money list; three spots shy of regaining his PGA Tour card.[10]
In 2016, Tway returned to the Web.com Tour and his results improved from the previous season. He scored four top-10s in 20 starts, including a solo third at the Price Cutter Charity Championship, but he still finished the regular season money list a few spots shy of automatically earning his card in 27th.[11] However, during the 2016 Web.com Tour Finals, Tway performed well, including a tie for third at the Nationwide Children's Hospital Championship, which earned him back his PGA Tour card for 2017.
In 2017, Tway's season began with a slow start without having a top-30 finish in his first eleven starts, but in April, Tway knocked off three consecutive top-5s, finishing T3 at the Valero Texas Open, solo 3rd at the Zurich Classic of New Orleans (partnering with Kelly Kraft), and T5 at the Wells Fargo Championship. Tway also moved from 175th in the FedEx Cup to 55th in those events, virtually locking up his Tour card for the next season. He made the playoffs for the first time and finished the season 69th in points, easily retaining his card for 2018.[12]
Tway had a solid 2018 season, making 26 of 33 cuts and scoring three top-10s. Again, Kevin had a great spring with T9 at the AT&T Byron Nelson, top-5 at the Fort Worth Invitational, and T6 at the Travelers Championship. He made the playoffs for a second consecutive year and finished 87th in points, again easily retaining his PGA Tour card for the next season.[13]
On October 7, 2018 (beginning of 2019 season), Tway won the Safeway Open, for his first career PGA Tour victory, in a playoff over Brandt Snedeker and Ryan Moore. Tway birdied the last two holes in regulation to join a three-man playoff. Tway and Moore both birdied the first playoff hole, however Snedeker made par, eliminating him from the playoff. The next hole, Tway and Moore both birdied, forcing a third playoff hole. On the third playoff hole, Tway won with a 10 foot birdie putt.[14]