Johnson, 69, won the National League honors after guiding the Nationals to a 98-win season and their first division title, engineering a major turnaround after taking over from Jim Riggleman in midway through the 2011 campaign.
The Nats' win total marked an 18-game improvement over the previous year and their NL East crown was the first for the franchise since 1981, when it was known as the Montreal Expos.
Johnson's Manager of the Year award was his second -- he also was chosen as the AL's top skipper in 1997 while with Baltimore.
Melvin, meanwhile, garnered the AL honors with another turnaround story in Oakland, where the A's improved by 20 games over 2011 and won an unexpected AL West title.
He guided them to an MLB-best 57-26 after June 30, including a season-ending three-game sweep of Texas to clinch the division.
Melvin, 51, also won the NL Manager of the Year while with the Arizona Diamondbacks in 2007.
Celebrity Families of 2014 [PHOTOS]