Marcos Leonardo scored his second of the night in the 113th minute as Simone Inzaghi's side saw off the Premier League club, who had taken a ninth minute lead through Bernardo Silva.
Quick-fire goals from Leonardo and Malcom in six minutes immediately after the restart put Al-Hilal in front before Erling Haaland equalised for City in the 55th minute.
Four minutes into extra-time, Kalidou Koulibaly headed in Ruben Neves's corner to put the big-spending Riyadh-based club back in front and while Phil Foden levelled again for City, Leonardo netted with seven minutes left to take Al-Hilal through.
Earlier, Brazilian side Fluminense stunned Champions League finalists Inter Milan by knocking them out with a 2-0 victory in Charlotte.
The reign of new Inter Milan coach Cristian Chivu, who took over following Simone Inzaghi's departure just days after that humbling loss to Paris St Germain a month ago, has not started well as they exited the tournament before the quarter-finals.
Fluminense struck after three minutes when Cano pounced on a deflected cross and headed in from close range, putting the ball in between goalkeeper Yann Sommer's legs for a dream start.
The Italians pushed for an equaliser after the break but were frustrated by a well-organised defence. Stefan de Vrij missed a close-range chance, keeper Fabio denied Sebastiano Esposito with his feet and Lautaro Martinez struck the post.
Substitute Hercules sealed the win for Fluminense three minutes into added time, calmly slotting home from the edge of the box to send the Brazilians through to a last-eight clash with Al Hilal on Friday in Orlando.
Fluminense played with more poise throughout the match and became the second Brazilian club team to reach the quarter-finals, joining Palmeiras.
Milan certainly had their chances, but couldn't will the ball into the net.
Lautaro Martinez had a point-blank shot in the second half to tie the game, but the ball bounced off the upright.
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