Jonathan De Guzman cancelled out Lorenzo Insigne's opener to put the Welsh side ahead on away goals but the prolific Argentine and the Swiss struck late on
After a goalless first leg at the Liberty Stadium, Lorenzo Insigne lofted the ball over Michel Vorm on 16 minutes before Jonathan De Guzman levelled to put the Welsh side ahead on away goals.
But Higuain capitalised on a defensive mix-up to hammer in the winner for the hosts on 78 minutes and bring an end to Swansea's first ever European campaign before Gokhan Inler struck a third at the death.
Victory means Benitez, who won the tournament last year while in charge of Chelsea, remains on course to defend his title.
Marvin Emnes, one of three Swansea changes to the starting XI from the home leg, had the first sight of goal in the 11th minute, but, after being played in by Pablo Hernandez, the Dutchman saw his chipped effort over Reina cleared from just in front of the goalline by Raul Albiol.
The hosts ought to have broken the deadlock themselves four minutes later, but Higuain blazed over from 10 yards after linking up with Goran Pandev.
Swansea failed to heed the warning in the 17th minute, Insigne springing the offside trap before lofting his finish over Vorm and into the net.
The Swansea defence was in disarray, Higuain flashing an effort a yard wide with the visitors apparently still reeling from falling behind.
The Premier League side recovered their composure, however, and were level on the half-hour mark.
A slick passing move resulted in the ball being fed into Wilfried Bony and his clever flick was picked up by De Guzman, who fired left footed beyond Reina to provoke jeers from the home crowd.
The goal gave Swansea added confidence and they enjoyed long spells of possession, Bony cutting inside Henrique before testing Reina as the half drew to a close.
Midway through the second half, Insigne broke through only to see his effort blocked by Chico Flores, before Bony wasted a glorious chance when he headed Hernandez's cross straight into the arms of Reina.
Swansea hearts were broken on 78 minutes when the ball bounced off both Flores and Davies and landed perfectly for Higuain, who made no mistake from 10 yards.
With almost the last kick of the game and with Swansea completely exposed, substitute Marek Hamsik fed Inler, who beat Vorm to seal the win and set up a tie with Porto in the next round.