Melvin, 57, edged out Boston Red Sox manager Alex Cora and Tampa Bay Rays skipper Kevin Cash in the American League. He received 18 first-place votes to beat out Cora, who received seven. The Athletics had a 22-win improvement from 2017 to 2018, going 97-65 to finish in second place in the AL West.
Melvin is the eighth manager in history to win his third Manager of the Year award. He previously took it home in 2007 with the Arizona Diamondbacks and 2012 with the A's.
Snitker, 63, beat out the Milwaukee Brewers' Craig Counsell and the Colorado Rockies' Bud Black. He received 17 first-place votes after leading the Braves to a 90-72 record and NL East championship. A longtime manager in the Braves' farm system, Snitker took over the big league club in 2016.
Legendary Braves manager Bobby Cox is the only other winner in franchise history.
"He's the first guy I called when I got this job, after the interim tag was removed," Snitker said of Cox on MLB Network.
Atlanta signed Snitker to a two-year contract in October.
"It does give you a sense of accomplishment, I guess," Snitker told reporters. "I feel really good about it. I feel good I have the opportunity to stay around longer and be a part of this."
Counsell led the Brewers to a 96-67 record and narrowly missed out on repeating as Manager of the Year. Milwaukee reached Game 7 of the National League Championship Series.
Black finished third after the Rockies went 91-72 and earned a wild-card berth. The Rockies were eliminated by Counsell's Brewers in the NLDS.
Cora led the Red Sox to an MLB-best 108-54 record and took home a World Series title in his first year as manager.
The Rays finished 90-72 under Cash this season despite having the second-lowest payroll in baseball.