Bale, who was left on crutches following a challenge from Liverpool midfielder Charlie Adam in Saturday's goalless draw against their Premier League rivals, started Tuesday night's game and headed home a 58th-minute equaliser to cancel out Kenny Cooper's early penalty for the hosts.
New signing Gylfi Sigurdsson then curled in a superb match-winning strike seven minutes after Bale's goal to give Andre Villas-Boas' men a 2-1 win at Red Bull Arena.
The Red Bulls took the lead in the eighth minute when Tim Cahill, making his debut for the Major League Soccer side following his move from Everton, went crashing in the box under a challenge from new Spurs defender Jan Vertonghen and Cooper slotted home the resulting spot-kick.
Spurs went in search of an equaliser before the break and Bale and Tom Huddlestone both drew fine saves from New York goalkeeper Bill Gaudette, who then produced a brilliant one-handed stop to tip over Aaron Lennon's effort in the 53rd minute.
Gaudette was helpless to prevent Bale drawing the teams level five minutes later though as the Wales winger climbed high above his marker to glance Sigurdsson's corner into the back of the net.
Sigurdsson then sealed the win with a fine effort, the Iceland international working his way into the box before cutting inside Markus Holgersson and curling an unstoppable shot into the far top corner.
Spurs also hit the crossbar through Michael Dawson following another Sigurdsson corner soon after as Villas-Boas men signed off their US tour with a deserved victory.