A solitary strike from Aiden McGeady sunk Benfica to keep alive Celtic's aspirations of reaching the last 16 of the Champions League.
Aiden McGeady gave Celtic the lead on the stroke of half-time
Oscar Cardozo was a constant threat for the visitors in an open first half, bringing out the best in Celtic goalkeeper Artur Boruc.
Scott McDonald went closest for Celtic before McGeady's deflected shot on the stroke of half-time flew past Quim.
The Scottish Champions assumed control after the break as Benfica faded.
And the hard-earned victory continued Celtic's impressive home record in Champions League competition with their unbeaten run stretching back to September 2004.
Benfica enjoyed most of the early possession and were moving the ball about with impressive fluency.
And the Portugese side forced Boruc into action after seven minutes when he denied Cardoza with a decent save from point-blank range after the striker was nicely teed up by Cristian Rodriguez.
The home side then spurned a glorious opportunity to open the scoring after the lively McGeady was hauled down by Rui Costa.
Paul Hartley sent a curling free-kick into the box only for John Kennedy to nod inches wide with Quim's goal at his mercy.
The action flowed from end to end and Cardozo was proving to be a handful for the Celtic defence.
The Argentinian pounced on a loose ball after 16 minutes and accelerated past Lee Naylor into the box before unleashing a rising volley that Boruc did well to turn over the bar.
At the other end Quim produced an impressive save of his own to push away a ferocious long-range drive from Scott Brown.
Despite Benfica's dominance in possession, Celtic created a flurry of great chances midway through the first half.
McGeady was in sparkling form for the Scottish Champions
McDonald was denied twice in quick succession by Quim and Brown swiped at fresh air six yards out with the Benfica defence in disarray.
But just when the half looked set to peter out goalless McGeady jinked in from the left and struck a sweet right-foot shot which took a wicked deflection off Luisao to give Celtic the lead.
The home side started the second half brightly and their high tempo game was unsettling the visitors.
McGeady and McDonald continually tormented the Benfica back-line with their energetic movement as Celtic went in search of the decisive second goal.
With Celtic in the ascendancy Cardoza blew a gilt-edged opportunity to snatch an equaliser for Benfica after 62 minutes when he was picked out in the box by Leo's floated cross.
But the striker's attempted header came off his shoulder and flew just past Boruc's left-hand post.
With 20 minutes to go Brown burst through the heart of the Benfica defence but Quim did well to block his thunderous drive and the Celtic midfielder could only nod the rebound over the bar.
Benfica were reduced to ten men five minutes from the end when Augustin Binya was red-carded for a shocking challenge on Brown.
The Scottish Champions controlled the remainder of the game without creating any more clear-cut chances and held on comfortably in the end to pick up a vital three points and stay in the hunt in Group D.
Celtic: Boruc, Naylor, Caldwell, Kennedy, McManus, Scott Brown (Sno 89), Hartley, Jarosik (Donati 66), McGeady, Vennegoor of Hesselink (Killen 66), McDonald. Subs Not Used: Mark Brown, Zurawski, Riordan, O'Dea.
Goals: McGeady 45.
Benfica: Quim, Luis Felipe, Edcarlos, Luisao, Leo, Katsouranis, Rui Costa (Bergessio 77), Pereira (Di Maria 61), Binya, Rodriguez, Cardozo (Nuno Gomes 77). Subs Not Used: Butt, Zoro, Nuno Assis, Adu.
Sent Off: Binya (85).
Ref: Martin Hansson (Sweden).