The Cards’ win, coupled with Elizabethton's 2-1 loss at Greeneville, left Johnson City up by two games with three to go in the Appalachian League West Division.
The Cards and Twins will face each other in the first-round playoff opener on Sept. 3.
With the Cardinals clinging to a 3-2 lead in the ninth, Bluefield's Bradley Jones led off with a blast to center field. The ball caromed off the wall back toward the infield.
Jones tried to stretch a triple into an inside-the-park homer, but Cards center fielder DeAndre Asbury-Heath picked the ball up and threw to the plate, where Garcia caught the ball and tagged Robinson out.
In the sixth inning, Erik Garcia drew a bases-loaded walk, scoring Hunter Newman with the go-ahead run and a 3-2 lead.
Garcia had given the Cards a 2-0 lead in the second inning with a two-run homer, his first of the season.
Johnson City starter Frederis Parra (3-5) worked seven innings, allowing six hits and two runs.
Keaton Siomkin pitched the eighth and Estarlin Arias pitched the ninth for his sixth save.
The Cardinals made the most of five hits. No player had more than one.
Jones and Kalik May had two hits apiece for the Blue Jays, fighting for the second playoff spot in the East Division.