The Spaniard broke the deadlock shortly before the break with a tap in from Eduardo Salvio and struck again in similar fashion with 19 minutes to go.
The Lisbon club - which dropped into the competition after elimination from the Champions League - was defeated 2-1 by Chelsea in the 2013 final, but moved a step closer to atoning for that disappointment as a double from Rodrigo secured a 3-0 aggregate success Thursday.
Benfica's misery last season was not confounded to the Europa League, losing the league title by a point following a last-minute defeat to champions Porto before succumbing to a shock defeat against Vitoria Guimaraes in the Taca de Portugal final.
Jorge Jesus' men are in a position to make amends for those setbacks though. They lead the Portuguese top flight by seven points with four games to go, while semifinals against Porto in the Taca de Portugal and Taca da Liga are still to be decided.
Rodrigo kept Benfica on course for an impressive quadruple, finding the net in each half to seal the hosts' place in the last four and condemn a meek AZ side to elimination.
Nicolas Gaitan and Maximiliano Pereira missed out through suspension for Benfica, who suffered an early blow inside two minutes as fullback Silvio sustained an apparent serious injury in a collision with teammate Luisao.
Both players were trying to tackle Roy Beerens but Silvio had to be taken off on a stretcher and was replaced by Andre Almeida.
The early blow did not prevent Benfica from making the better start, with only a flying save from AZ's Costa Rican goalkeeper Esteban Alvarado keeping out Oscar Cardozo's curling strike from the edge of the penalty area.
Chances proved few and far between in a largely disappointing first half, but Cardozo was presented with a glorious opportunity just after the half-hour mark, only to see his left-footed effort superbly turned behind by Alvarado after Rodrigo's cut back.
Paraguay international Cardozo then fired wide following more good work from Rodrigo, who finally broke the deadlock six minutes before the interval, popping up at the far post to slot home Eduardo Salvio's excellent right-wing cross from close range.
AZ finally crafted a chance midway through the second half, with defender Mattias Johansson forcing Benfica goalkeeper Artur Moraes into action with a long-range drive.
But any hopes of a fight back from the visitors were killed off in the 71st minute, when Rodrigo turned home another fine Salvio delivery with a well-executed volley.
Benfica nearly added a third in the closing stages, with Salvio – scorer of the winner in the first leg – denied a deserved goal by Alvarado.