But Madrid will progress to the semi-finals to face Chelsea thanks to the 3-1 aggregate scoreline against Liverpool.
The Madrid boss had the last laugh as he was grinning and smiling in the background of Casemiro and Robertson's altercation.
Casemiro avoided a potential red card after his late sliding tackle on James Milner down by the touchline.
Liverpool boss Jurgen Klopp was furious as the 29-year-old Brazilian midfielder received a yellow card.
VAR was checked and Casemiro was fortunate not to be sent off, but the Madrid star was also involved in a mini-clash with Robertson.
Robertson was booked for his part in the heated confrontation with Casemiro, but Madrid fans couldn't help but laugh at Zidane's reaction in the background.
Speaking after the 0-0 draw, Milner insisted that Liverpool were the better side on the night and the first leg would have been "very different" if they had played with the same intensity.
The 35-year-old Liverpool vice-captain told BT Sport: "The performance, intensity and desire was good. But you can't not take the chances we had and that was the difference.
"The positive is we are creating the chances and the reaction we showed. We were obviously the better team. We did enough to win the game.
"We wanted to come out of the blocks and set that tempo. If we had played that first leg with the same intensity, it would have been very different. Ultimately, we weren't good enough over two games.
"The most important [part] is if we perform with the desire and intensity, hopefully we can get into those top-four spots."
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