Benfica came into the fixture four points clear of local rivals Sporting Lisbon, who sit second, but now hold a seven-point advantage with just three matches left to play.
Rodrigo's fourth goal in his last three matches put the visitors ahead in first-half stoppage time after the Arouca defence failed to clear their lines from a Lima cross.
In Sunday's other match, Porto needed goals from Carlos Eduardo and Juan Quintero in the last four minutes to give them a 3-1 win at Sporting Braga.
Brazilian Eduardo restored Porto's advantage four minutes from time though, and Quintero's stoppage-time effort sealed the points.
Victory lifts Porto nine points clear of the fourth-placed Estoril, who face Pacos de Ferreira tomorrow, while Braga's hopes of European football next season are fading.
