The Paraguayan striker had twice been denied woodwork in the second half, but produced a cool finish with four minutes left to give Benfica their first points in the group.
It was cruel luck on Celtic who were disciplined and organised in defence but their luck finally deserted them.
Augustin Binyi tried his luck from distance early on for Benfica but his shot was well wide of Artur Boruc's left-hand post.
Celtic responded swiftly and a promising break from Aiden McGeady set up Jiri Jarosik but his shot was deflected wide for a corner that came to nothing.
McGeady put his side in trouble midway through the first half when he was short with his headed pass back but Lee Naylor managed to swipe the ball away before Benfica could take any advantage with Cardozo looking to cash in.
Celtic had a rare chance when Chris Killen headed into the area and Massimo Donati managed to pick up the loose ball, but he did not get hold of his shot properly and Silva Quim managed to push it past his post.
Jarosik then set up Scott Brown but his shot was narrowly wide of the post with Quim exposed.
It took a fingertip save from Boruc to keep out a header from Luisao on the stroke of half-time as Celtic lived dangerously.
Benfica started the second half strongly and it took another stunning save from Boruc to keep out a header from Cardozo. Rui Costa then fired carelessly over after some neat skill.
But Celtic broke and Killen hit the sidenetting after more good work from McGeady as the game finally caught fire.
A brilliant solo run from McGeady almost brought the opener but Quim was equal to his shot and turned it past for a corner.
There was a major let-off for Celtic after 63 minutes when Cardozo had the goal at his mercy but thumped his shot against the underside of the crossbar after Boruc had parried an effort from Christian Rodriguez.
Moments later, the Paraguayan had another chance and beat Boruc again but this time his shot hit the outside of the post.
But it finally came good for Cardozo when he beat the Celtic offside trap to score the winning goal with just four minutes left.