Cuevas lined a one-out double to left, plating Naoya Washiya and Eury Perez to give the Nationals a two-game sweep of the best-of-3 finals. Right-hander Wilson Eusebio (1-0) struck out five over four innings of relief to earn the win.
The championship comes a year after the Nationals lost to the Phillies in the finals.
"This is a special win," said Nationals manager Bob Henley. "There's a lot of guys on this team who were here last year when the Phillies celebrated at our stadium, so it's very sweet for them, and definitely sweet for me."
The Nationals, who finished the season 36-19 to earn the wild card, trailed, 3-0, in the fourth inning but clawed back despite being out-hit 12-10.
"They're an extremely scrappy, scrappy bunch. They continued to find a way to win," said Henley. "We got down early but never felt out of it."
Angelberth Montilla's solo homer in the fifth tied the game at 3-3, but the Marlins regained the lead an inning later when Wilfredo Gimenez scored on Eusebio's wild pitch.
Gimenez hit an RBI double and scored on Jobduan Morales' run-scoring single as the Marlins scored twice in the top of the fourth.
The Nats scored a pair of runs in the bottom of the frame when Cuevas raced home on a wild pitch and Adrian Nieto plated Brett Newsome with an RBI single.
Newsome, who went 1-for-4, said the team's experience in last year's championship gave them an edge this September.
"It gave us a lot of confidence going in," Newsome said. "Some of the players from last year are still on the team this year. We got to the championship game last year, won the first game and then lost the next two, so we knew we couldn't let down. They scored in the first inning and we just had to battle back."
The Marlins led the GCL with 38 wins and took the East Division crown before rolling past the Yankees in the first round. Newsome said the Nationals, who took Game 1 on Wednesday, 3-1, were prepared for the challenge.
"We knew it was going to be between us and the Marlins," said Newsome. "They had the best record, and going in, we knew it could be a possibility. We saw good pitching all playoffs long, and our pitching did most of the work."
Starter Graham Hicks, Washington's fourth-round pick in the 2008 Draft, held the Marlins to three runs on seven hits over four innings. Eusebio bridged the gap to Rob Wort, who picked up his second save of the season despite allowing one hit in the ninth.
Marlins starter Saul Gonzalez allowed two runs over 3 2/3 innings before Jordan Conley (0-1) surrendered a pair of runs in three frames to suffer the loss. Gimenez finished 3-for-4 while Terrence Dayleg and Morales each drove in a run for the Florida affiliate.
Newsome, who signed with Washington as a non-drafted free agent out of Western Michigan University, spent 23 games in the GCL last summer. The 23-year-old is now one of the club's veteran leaders.
"I tried to keep everyone calm, everyone gets excited, there's lot of young guys on the team. I'm like the Crash Davis on the team," joked Newsome. "Now we have to go throw some grape juice on ourselves, not everyone's of age."
Danny Wild is an editor for MLB.com. Kristen Zimmerman contributed to this story. This story was not subject to the approval of Major League Baseball or its clubs.