It was the second consecutive win for the Royals, who snapped a 12-game losing streak Wednesday.
Mendoza (1-2) gave up four hits while walking two and striking out three over five innings. Tim Collins and Aaron Crow each held the 4-2 Kansas City lead before Jonathan Broxton, despite allowing a single and a walk in the ninth, picked up his second save.
In addition to the six hits -- all singles -- the Royals gave up five walks and had a total of six strikeouts.
Cleveland starter Josh Tomlin (1-2) allowed four runs and eight hits in 4 2/3 innings.
Kansas City opened the scoring in the first with Yuniesky Betancourt scoring on a double play. The Indians tied the score in the third on an RBI single from Aaron Cunningham.