Despite taking a 1-0 lead midway through the second half, the Wanderers had skipper Nikolai Topor-Stanley and Matthew Spiranovic sent off as the Mexicans won in extra-time.
A penalty two minutes from time allowed Cruz Azul to equalise before Mariano Pavone and a second spot-kick from skipper Gerado Torrado sent them into a semi-final clash with Real Madrid.
"That whole 90 minutes to me was disgusting, absolutely farcical what happened out there and totally unacceptable," Covic told Fox Sports.
"The conditions were hard enough as it is and we’ve copped decisions ultimately that gave them the game.
"There’s no other way to talk about it. It was never going to be a pretty game in the conditions but to me it’s embarrassing."
Torrential rain saw the match played in atrocious conditions, with puddles forming all over the pitch and making it almost impossible to play crisp passes or dribble with the ball.
The performance of Ivorian referee Noumandiez Doue did not improve the Wanderers’ mood either, especially the controversial dismissal of Topor-Stanley in extra time.
A foul on Pavone saw Topor-Stanley booked in regulation time, even though the defender was nowhere near the incident in a clear case of mistaken identity.
"As far as we know it should have been a yellow to (Seyi) Adeleke in the first place and they’ve got it down as Nikolai and given him a second yellow card," Covic said of the incident.
"The sending off was fairly soft, what more can you say?"
Asked if he felt the game should have been abandoned due to the waterlogged pitch, Covic said: 'You can’t play football on that.
"You can have as much skill as you want but you can’t play football on that. We came in at half-time and didn’t even have a team talk, we just wrote ‘play the conditions’ and that was it. It was never going to be pretty."
