Kelvin Herrera gave up the game-tying single to David Freese before Chris Iannetta launched the go-ahead home run off Tim Collins in the eighth as the Angels came back to win 4-3 in the series finale with Kansas City.
Every Royals (24-24) run came in the third, beginning with Pedro Ciricaco's RBI double to score Jimmy Paredes. Eric Hosmer plated one on a sacrifice fly, while Alex Gordon drove in Jarrod Dyson with single.
Vargas worked 6 1/3 innings, allowing just three hits and five walks while striking out six.
Angels (28-21) starter Garrett Richards struck out seven, but surrendered three runs on five hits.
The Royals return for a quick three-game homestand against the Astros beginning Monday.