Tigre led 3-1 with 27 minutes remaining in the rescheduled Argentina Primera Division round eight fixture on Wednesday.
But hosts Godoy Cruz - aided by Erik Godoy's red card in the 72nd minute - hit back with two unanswered goals to level proceedings at the Estadio Malvinas Argentinas.
And Uruguayan midfielder Rodriguez completed the comeback in the 91st minute, firing a thunderous long-range shot into the roof of the net, as Godoy snapped a six-game losing streak.
Godoy Cruz have collected 13 points from 11 matches, three less than ninth-placed Tigre.
Tigre took a 17th-minute lead through striker Carlos Luna, who was played in by team-mate Sebastian Rincon before looping the ball over Godoy Cruz goalkeeper Sebastian Moyano.
Ecuador striker Jaime Ayovi equalized after 30 minutes, heading home from a corner for the home side.
But Luna netted his second of the match two minutes later as Tigre took a lead into the break.
Leandro Garate come off the bench and doubled Tigre's advantage just past the hour-mark, beating two defenders and placing the ball beyond Moyano.
Ayovi kept Godoy Cruz in the contest with a diving header inside the six-yard box in the 69th minute.
Godoy Cruz's hopes were boosted in the 72nd minute, when full-back Godoy saw red.
And the hosts took full advantage, Gonzalo Cabrera levelling the scores four minutes later before Rodriguez produced the goods with a sensational effort from outside the box.
