Trout finished second in MVP voting both of his first two seasons, losing to Detroit's Miguel Cabrera. He joins Mickey Mantle as the only players who finished as MVP runner-up in consecutive seasons before winning in the following season.
Despite being a two-time runner-up for the award, Trout told MLB Network the possibility of winning his first MVP didn't enter his mind until after the season.
"At the end of the year, after the season gets over, you kinda look at your numbers and you think about it a little bit," Trout told the network.
Trout, 23, is the fifth-youngest winner of the award and the youngest to win in a unanimous vote.
He was tops in the AL with 111 RBIs this season and batted .287 with a career-high 36 home runs. He led the league in both runs scored (115) and total bases (338). He is the first AL MVP to win by unanimous vote since Ken Griffey Jr. in 1997.
"If you would've told me this before the season started, I would've just laughed at you," Trout told MLB Network. "It's just an unbelievable feeling."
He also led all position players in wins above replacement this season at 7.9, which was the lowest WAR in the standout outfielder's three-year career.
"Mike has had an incredible start to his career," manager Mike Scioscia said in a statement. "His play this year totally embodies what an MVP is all about. His terrific performance, along with his selfless style of play, has made him a tremendous leader on this team."
Trout was also honored this season with his third straight Silver Slugger award and was also named the MVP of the 2014 All-Star Game.
The Angels, who lost to Kansas City in the AL Division Series, finished 98-64 this season. Trout joins Vladimir Guerrero (2004) and Don Baylor (1979) as the only players in franchise history to be named MVP.
Trout received all 30 first-place votes and 420 points. Detroit designated hitter Victor Martinez was second with 229 points and Cleveland outfielder Michael Brantley third with 185.