Salvador Perez had the go-ahead RBI double for Kansas City in the bottom of the eighth and Greg Holland recorded his second consecutive save striking out two batters in the top of the ninth.
The Royals (2-2) have now won two in a row, and play for the series sweep on Sunday.
Chen struck out seven batters in his first start of the season, and left the game with Kansas City ahead 3-1. But the bullpen couldn't hang on to the lead.
Wade Davis allowed two earned runs in one inning for his first blown save of the season, but Perez bailed him out with the go-ahead hit in the next frame.
Billy Butler was 1-2 in the win with a pair of RBI. Perez lone hit was the game-winner. His batting average through four games is .538.
Kansas City sends James Shields to the mound Sunday as the Royals close their series with the White Sox.