The defeat was United's sixth in their last 10 games in all competitions -- a torrid run of form has left them bottom of Champions League Group A and 10th in the Premier League.
"Last season was brilliant, terrific, more than we can expect, but also in this project there would be gaps and this moment is a very difficult period," Ten Hag told a postmatch news conference.
"We are fighting together, we will stick together, we are behind each other; me, the directors and team all together, we will fight. We know we will do better and together we will come out.
"I am responsible for this team, this result, together with my squad. We have to do better but also I have seen a team with great spirit, and also one backed by the crowd from the start to the end. In the end the fans are disappointed as we are. But the good thing from this team is every time they find the energy."
Goalkeeper André Onana made a sloppy pass to Galatasaray forward Mauro Icardi that led to Casemiro giving away a penalty and being sent off. The error comes after a rough start to life at Old Trafford, coming after a mistake in their previous Champions League defeat allowed Bayern Munich to open the scoring.
"We are happy with our goalkeeping group," Ten Hag said. "Last season, Andre was in a final in the Champions League and can be one of the best goalkeepers in the world. We have seen already his great capabilities and also his personality. He will make a mistake and bounce back, and will do so again, I am sure," he said.
"There are no excuses. Maybe we're unbalanced [on the] left side but we still can't make the errors we are and we have to do better, simple as that."