The Portuguese winger claimed the decisive goal to keep Julen Lopetegui's side top of their Champions League group and on course for qualification
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, butGuillermo Fernandez's first goal since February brought Ernesto Valverde's men level.
Quaresma eventually grabbed the decisive strike, cutting in from the left-hand side and firing in an effort that goalkeeper Gorka Iraizoz failed to keep out.
Winless Athletic remain rooted to the bottom of the group ahead of the return fixture at the San Mames Stadium next month.
Porto showed the most attacking intent early on without initially managing to get behind the Spanish side's 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 denied a penalty when Bruno Martins Indi clattered into Aritz Aduriz with 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 Athletic'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 then 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.
Athletic 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 a draw looking likely, 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 as Porto held out to extend their advantage at the summit.