The midfielder doubled the New Zealand champions' lead when his blistering 30-yard strike found the back of the net at the Mohammad Bin Zayed Stadium in Abu Dhabi.
"Chad's goal was just class," Auckland coach Paul Posa told a news conference. "Chad Coombes is someone I feel is just starting to blossom."
Al Ahli had more chances in a scrappy first half until English striker Adam Dickinson latched on to a fine cross from Jason Hayne on the right wing and shot from close range just before halftime to give Auckland the lead.
Al Ahli had the chance to level the game less than a minute later but captain Salem Khamis' shot from 10 yards out sailed over the crossbar.
Brazilian striker Bare went close for Al Ahli in the second half but Auckland survived late pressure to progress to the quarter-finals, where they will meet Atlante of Mexico.
"The key for us was the homework we did on Al Ahli. We knew how they were going to play so we had a system to counter that," Posa said. "For us this was a cup final."
Oceania champions Auckland play in the eight-team amateur New Zealand Football Championship, which began in 2004. They have won the domestic title four times.
"If we had scored an equalising goal in the first half, that would have given us a good advantage," Al Ahli coach Mahdi Redha said. "The team today really didn't play very well."
The tournament, which runs until Dec. 19, brings together the champions of FIFA's six confederations as well as host team Al Ahli.
European champions Barcelona and South American champions Estudiantes have byes into the semi-finals.
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