Gerrard - who came off the bench at half-time - saw red for a stamp on United midfielder Ander Herrera in his final game against Liverpool's biggest rivals before leaving for LA Galaxy.
The hosts were trailing 1-0 through Juan Mata's strike when Gerrard was dismissed, and fell 2-0 behind in the 59th minute through a spectacular scissor-kick from the Spaniard.
Daniel Sturridge halved the deficit, but a comeback never materialised, with United forward Wayne Rooney seeing an injury-time penalty saved by Simon Mignolet after Emre Can had felled Daley Blind.
Rodgers told Sky Sports: "Steven apologised to the players and me in the dressing room. It was probably frustration, we didn't put a tackle in in the first half but the players adjusted well.
"I was disappointed with the first half, our start position was way too deep. They were excellent and deserved their lead. But the players showed incredible bravery in the second half and showed character.
"Without playing great, we still had a good chance. We needed some luck but it wasn't to be. We will get a reaction after this and finish the season strongly."
Ugly scenes somewhat marred the end of an enthralling game after Martin Skrtel appeared to stamp on United goalkeeper David de Gea.
"I have seen the Martin Skrtel incident," Rodgers added. "The ball was played through but it probably looks worse in slow motion.
"It looks like he is trying to get a nick on the ball, to get a toe on it the ball went into the goalkeeper's hands. Martin is not that type of player, he was just a fraction out.
"We have to refocus on our next game and we knew there would be one or two defeats on reaching our points total."
