Mazzeo rose at the far post to meet Marcus Younis' inviting cross from the right, the 20-year-old outjumping the defence to power a downward header beyond Buriram goalkeeper Neil Etheridge to keep City's hopes of a place in April's quarter-finals alive.
"It's certainly important that we scored because if we went there 1-0 down, there would probably be a little bit of a lack of belief," said City coach Aurelio Vidmar. "But I thought we thoroughly deserved what we got tonight."
Buriram, who are looking to reach the last eight for the second season in a row, took the lead through Guilherme Bissoli, who rifled a stunning first-time strike into the top corner from a tight angle in the 37th minute.
The Brazilian was awarded a late penalty by referee Khaled Al-Turais, only for the decision to be overturned by VAR officials and, and moments later, Mazzeo headed in the equaliser.
Japan's Machida Zelvia held on for a 0-0 draw with Gangwon FC in South Korea after goalkeeper Kosei Tani denied the hosts late in the game.
The Japan international threw himself at full stretch to his left to push substitute Abdalla Halaihal's goal-bound strike onto the post in the 88th minute and the teams will meet again in Machida next Tuesday.
Japan's Vissel Kobe will take on FC Seoul from South Korea on Wednesday, with Johor Darul Ta'zim from Malaysia playing J-League outfit Sanfrecce Hiroshima.
The winners of the last-16 ties from both west and east Asia will advance to the next phase, in which the quarter-finals, semi-finals and final will be played on a centralised basis in Jeddah in April.
The Asian Football Confederation announced on Sunday that matches in the western side of the draw have been postponed until further notice due to the crisis in the Middle East.