The Cubs escaped a jam in the top of the 14th and then got to Jonathan Broxton in the bottom half of the inning to salvage the finale of a four-game series.
Castro reached on a single off Jonathan (2-2) to open the 14th, stole second and took third on a ground out by Alfonso Soriano.
Nate Schierholtz was walked intentionally to set up the possibility of a double play, but Borbon, hitting for pitcher Hector Rondon, sent a 1-0 pitch into left field for the game-winning RBI.
Chicago trailed 5-3 heading into the sixth inning, but Schierholtz scored in that frame on a ground out by Luis Valbuena and Schierholtz again came in with the tying run in the eighth on a single by Darwin Barney.
Rondon (1-0), Chicago's seventh pitcher of the game, picked up the first victory of his big league career by working the 13th and 14th innings. This was his 20th relief appearance over parts of three major league seasons with Cleveland and Chicago.