Anderson missed their Europa League win against Porto after the passing of his mother this week. The 1-0 win saw Forest reach the semifinals 2-1 on aggregate.
Gibbs-White said afterwards: "Really emotional night. It's a big shame, he's an incredible kid and no one deserves this but we're all with him."
Chants of "Istanbul" rang out at the City Ground near the end with the final to be staged in Turkey this season.
"They were chanting that four games ago,” Gibbs-White smiled. “Their expectations are high and so are ours. Villa in the next round, we take it as it comes but we have a Premier League game to focus on first."
He added: "We got fortunate (with) them going down to 10 men (Jan Bednarek), we knew it would be difficult. Incredible win and we go onto the semi-final. An incredible night.
"One of the best nights I've had at the City Ground, thanks to the fans and everyone that's made it happen. We said early on that we'd take this competition full on and we deservedly got the win today. Now we rest and go into the league."