The Iranian side had appeared to be headed for a victory that would have sent them through to the knockout round with two games to spare when Ali Karimi scored with a deflected shot six minutes from time.
But despite being a player down for the final 19 minutes after the dismissal of their Iranian striker Farhad Majidi, the Qatari club drew level when Quaye launched a screamer from 20 metres into the top right corner of the net.
Having seen his team trounced 3-0 at home by Piroozi just two weeks ago, Al Gharafa coach Paulo Silas was delighted with the resilience displayed by his players in the intimidating atmosphere of the Azadi Stadium.
"We achieved an honour by earning a point at the Azadi Stadium. It is also an important achievement for Qatari football," said the Brazilian.
"We played nice football and I'd like to thank Piroozi for having such understanding fans.
"It was a balanced game between two sides but Piroozi had most of the possession and that could be because of their experienced star, Ali Karimi."
While Piroozi still lead the group and should be able to wrap up their place in the knockout round in their remaining two games, coach Mustafa Denizli was disappointed that his players did not make better use of their opportunities.
"At times, our players were influenced by the crowd and they played emotionally when we wanted to score quickly," said the Turk.
"This draw has disappointed all of us because we wanted to have 10 points after this match. "
"We wanted to play an attacking game and we pushed forward in the second half but we didn't take advantage of our chances. We also didn't use our free-kicks well enough.
"But we are still in charge of our own destiny as far as qualification for the next round is concerned and we are not going to concern ourselves with other results," he added.
Al Gharafa suffered an early blow in the game when striker Aruna Dindane had to be replaced in the 14th minute after he had appeared to roll his ankle.
The first half was a rather lifeless affair with the first real scoring opportunities only arriving close to the half-hour mark when Piroozi goalkeeper Asmir Avdukic made good saves to keep out a Farhad Majidi header and Quaye's acrobatic kick.
Mohammad Nori had the Iranian side's best chance of the half in the 33rd minute as he poked a cross from the right towards the net but saw his effort deflected wide for a corner.
Piroozi took control early in the second half with Ali Karimi shooting over from the edge of the box before the Reds skipper crossed for an unmarked Eamon Zayed, who headed wastefully over from point blank range.
Al Gharafa had chances of their own and Ibrahim Al Ghanim nearly put them in front when he met Fahid Al Shammari's cross from the right with a glancing header that went narrowly wide.
Majidi was booed constantly by the Piroozi fans because of his past connections with their bitter rivals Esteghlal and he raised perhaps the loudest cheers of the night when he was sent off in the 72nd minute after he received his second yellow card in five minutes for what the referee deemed to be simulation when he went down in the Piroozi box following a challenge by Mohammad Nosrati.
Piroozi laid siege to the Al Gharafa goal after that but goalkeeper Qassim Borhan did brilliantly to deny Amir Hossein Feshangchi and Mohammd Nori in quick succession.
But in the 84th minute, Karimi broke the deadlock when he sidestepped challenges by Al Ghanim and Hamed Shami in the Al Gharafa box before taking a shot which took a wicked deflection off Otman El Assas and looped past Qassim Borhan.
However it was not the end of the story as the visitors levelled in added time when Diego Tardelli laid the ball off for Quaye, who launched a fierce shot that took a slight nick off a defender on its way into the net.