The visitors were on the back foot early and, while attempting to clear inside his box, Dane Murray instead caught Jhon Duran and a spot kick was awarded after just five minutes.
Duran dusted himself down and coolly dinked the ball down the middle to open the scoring.
To their credit, Martin O’Neill’s men responded well initially with Mika Baur forcing a smart stop from Anatoliy Trubin just two minutes later, but the respite was short-lived as Benfica pressed forward again.
Gianluca Prestianni curled over the bar from inside the box on 20 minutes but, just two minutes later, would turn provider for Alexander Bah, who was given the freedom of the penalty area to side-foot home on the volley at the back post.
Once again, the visitors responded with a chance of their own, and they rattled the woodwork twice in quick succession through Kasper Hogh and Camilo Duran on the rebound.
Despite coming close to getting back into the game, the Scottish champions were getting exposed at the back at an alarming rate, and Duran took advantage with just under 10 minutes until the break.
It was Prestianni who was the provider again, finding Duran, who held off the challenge of Auston Trusty and fired a shot through the legs of Viljami Sinisalo from a narrow angle.
O’Neill made a pair of changes at the break, and both made impacts within 10 minutes of the restart.
Haissem Hassan beat his man down the right with a nice bit of skill, but his shot left something to be desired, while Callum Osmand stabbed over at the back post.
The travelling support were given something to shout about soon after, however, as more positive play from Hassan allowed him to pick out Duran with a cross and the Colombian nodded home.
With three of the four goals now scored by a Duran, Camilo almost joined his namesake Jhon with a double, only to rattle the woodwork again with just under 20 minutes remaining.
Celtic still looked vulnerable at the back, and they had the post to thank this time as, just moments after Duran’s effort, sub Andreas Schjeldrup smashed his effort off Sinisalo’s right-hand post.
Despite their improved second-half performance, the away side had given them too much to do in the first half, and there was still time for sub Vangelis Pavlidis to add a late fourth.
The Celts will look to stop the rot of three consecutive European defeats as they welcome Celta Vigo next week, while Benfica will look to win their third consecutive Europa League game in Cyprus against Omonia in seven days’ time.
Flashscore Man of the Match: Gianluca Prestianni (Benfica)