The experienced winger struck 15 minutes from time at the Estadio Do Dragao on Tuesday to keep the hosts two points clear of Shakhtar Donetsk, who moved into second with a 7-0 trouncing of BATE.
Julen Lopetegui's side created the better chances early on and were rewarded when Hector Herrera opened the scoring shortly before the interval.
The Mexico international coolly fired home after a sublime ball from Juan Quintero, but Guillermo Fernandez's first goal since February brought Ernesto Valverde's men level.
Quaresma eventually grabbed the decisive strike for Porto, cutting in from the left-hand side and firing in an effort that Gorka Iraizoz should perhaps have done better with.
The visiting goalkeeper was unable to keep the shot out and winless Bilbao remain rooted to the bottom of the group ahead of the return fixture at the San Mames Stadium early next month.
In a lively opening from both sides, it was Porto who showed the most attacking intent early on without initially managing to get behind the Bilbao defence.
It took 16 minutes for a first effort on target to arrive as Quintero's curled strike from distance was collected by Iraizoz.
Bilbao were perhaps harshly denied a penalty when Bruno Martins Indi clattered into Aritz Aduriz - referee Damir Skomina instead awarding a free-kick for a foul on the Netherlands international.
Cristian Tello dragged a shot wide for Porto seven minutes before the half-hour mark but Bilbao's Mikel San Jose came closest in a first half of few chances when he rattled Fabiano's right-hand post with a deflected strike from outside the area.
The Porto goalkeeper gave coach Lopetegui a minor scare when he flapped at Markel Susaeta's delivery, but the hosts claimed the lead shortly before the break courtesy of Herrera's swept finish.
After taking care to remain onside, he had time and space to dispatch Quintero's clever pass and put his side in front.
Bilbao pulled level 13 minutes into the second half when Fernandez got onto the end of Benat's incisive pass before cutting inside Maicon and beating Fabiano to restore parity.
With Bilbao on course to salvage a draw, Quaresma was Porto's hero late on as he registered his first Champions League goal since scoring against Besiktas in December 2007 during his first spell with the club.
Fabiano was required to keep out Aymeric Laporte's header late on, but Quaresma's strike proved crucial as Porto extended their advantage at the summit.
