Lewis’ early opener silenced the 35,000 Forest fans behind the goal and Nuno Espirito Santo’s side failed to land a blow on City before the break.
Substitute Anthony Elanga should have made it 1-1, firing wide a minute after coming on at the start of the second half, but Gvardiol’s 51st-minute header put the game beyond reach.
Gibbs-White smacked the bar from the edge of the area with an audacious volley, before hitting the post from a tight angle after rounding City keeper Stefan Ortega following Gvardiol’s mistake.
Taiwo Awoniyi also hit the woodwork with an instinctive effort following more Forest pressure.
“We started really badly again, it makes everything hard. It took time to feel comfortable and adjust,” said Nuno.
“The first half was difficult and in the second half the momentum changed a little bit. We had a couple of chances which could put the game in a different story. The boys gave it all, we have to accept we lost.”
Forest next host Brentford in the Premier League on Thursday, sitting sixth in the table but knowing victory will return them to third place and maintain their push for a Champions League spot.
“It’s a sad day,” added Nuno. “It’s going to be hard, but when we wake up and the sadness is gone we know we have big things ahead of us to fight for.
“It can only bring us more energy for the last moments of the season.”
Midfielder Gibbs-White also looked to lift Forest, who beat relegation on the final day of last season.
He said: “We’ve got to put this behind us now and focus on the league. Five big finals, we are going to give it our absolute everything and hope for the best.”
