Portugal winger Quaresma crashed the ball in from the spot in the 78th minute after Nikola Maksimovic handled. Visiting captain Hamsik then levelled four minutes later with a rasping drive from outside the area.
Napoli dominated possession and could have avenged their 3-2 defeat by Besiktas two weeks ago but Lorenzo Insigne and substitute Dries Mertens fluffed chances in injury time.
The Italians have seven points from four games in Group B, level with Benfica who beat Dynamo Kiev 1-0.
Besiktas are in third place on six points while Kiev are bottom with one.
Napoli goalkeeper Reina felt his side deserved more than a draw.
"There was only one team on the pitch from start to finish," Reina said. "After conceding the goal we showed so much grit to fight back.
"This is the Napoli we all want to see. We allowed Besiktas very little and playing in Istanbul in the Champions League, that's not easy."
MILIK OUT
Napoli have looked lightweight up front in recent weeks without injured top scorer Arkadiusz Milik and their lack of a clinical finisher was underlined when Manolo Gabbiadini failed to beat goalkeeper Fabri twice from close range.
Besiktas were gifted a glaring chance when Kalidou Koulibaly's poor clearance set up Atiba Hutchinson but the Canadian hit the post.
Bolstered by the addition of Mertens from the bench, Napoli stepped it up in the second half.
The Turks, however, took the lead when Vincent Aboubakar's shot was blocked by Maksimovic's hand and referee Mark Clattenburg pointed to the spot.
Quaresma thumped the ball beyond Reina's reach to delight the boisterous home supporters.
Hamsik, though, quickly silenced them with an equally powerful effort at the other end.
"We gave our heart and soul to the end to get a result and looking at the statistics we did deserve more," said the Slovakia international.
Napoli have not gone beyond the group stages since 2012 but remain in a good position with two games remaining. They host Dynamo on Nov. 23 before travelling to Benfica.
"It was a game we could’ve won but considering the circumstances the point is worth its weight in gold," added Hamsik.
Besiktas next entertain Benfica on Nov. 23.
(Editing by Tony Jimenez)
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