Jason Vargas was dealing for seven innings only allowing one earned run on five hits to go along with six strikeouts.
Vargas' only blemish was Kinsler's solo home run in the fourth inning to give the Tigers a 1-0 lead.
Kansas City (0-2) didn't answer Detroit until the top of the ninth inning. With the bases loaded and one out, Alex Gordon hit a sacrifice fly to left field bringing in Pedro Ciriaco to tie the game.
Tim Collins entered the game in the tenth for the Royals in relief of Wade Davis. Collins walked two batters and recorded two outs before Kinsler delivered the walk-off RBI single.
The Royals bullpen's now lost each of the first two games of the season after Wade Davis took the loss on opening day.
Omar Infante had two of Kansas City's six hits in the game. Eric Hosmer also reached base twice going 1-3 with a walk. Salvador Perez had a hit in three at-bats to keep his batting average at a sky-high .714 just a pair of games into the season.
Kansas City and Detroit play the series finale Thursday at 12:08pm. Yordano Ventura gets the start for the Royals.