Having done the hard work, Frankfurt were left to thank goalkeeper Kevin Trapp for a brilliant fingertip save to deflect a late minute strike from Salvio on to a post.
But Frankfurt were deserving of their victory, having largely dominated the first period against opponents who seemed unsure about how to go about defending their 4-2 first-leg advantage.
Kostic forced the first save out of Benfica keeper Odisseas Vlachodimos after just eight minutes and the Croatian also came close from a corner as Frankfurt pressed.
Ante Rebic and Simon Falette also had chances for the hosts as Frankfurt's pressure pinned the visitors back deep in their own half.
Luka Jovic, causing plenty of problems down the flank, hit a long-range effort straight at Vlachodimos just past the half-hour mark, before Benfica's resistance finally crumbled five minutes later.
Mijat Gacinovic's initial effort rebounded and Kostic reacted quickest to slot it past the keeper, despite strong Benfica claims that the goalscorer had been offside in the build-up.
Benfica coach Bruno Lage was sent to the stands for protesting and it could have got worse for the visitors as Jovic headed just over the bar from a Kostic cross just before half-time.
Benfica began the second half looking brighter and Haris Seferovic should have converted a Felix cross shortly after the re-start.
Seferovic also had a poor shot easily saved by Trapp before Frankfurt rode the storm and regained the upper hand, Gacinovic shooting just wide them Rebic heading an effort over the bar.
Frankfurt grabbed their second goal after 67 minutes when the visitors failed to clear and the ball fell to Rode on the edge of the box and he drove it back past a helpless Vlachodimos.
Benfica could have salvaged their night late on when Felix headed on to the roof of the net and then Trapp summoned the excellent last-gasp saved which sent Frankfurt through.