The Blaugrana went ahead inside the opening eleven minutes thanks to the in-form Ansu Fati before Clement Lenglet was given his marching orders with around three minutes left to spare of the first half, having been cautioned twice.
Koeman’s men had to play with 10 men for the rest of the game but they still scored twice in the second half through Lucas Olaza’s own goal and Sergi Roberto’s 90th minute strike.
The Catalans have won a La Liga game at Celta Vigo for the first time in five years.
In an interview with the media after the game, Koeman admitted that Lenglet’s red card was harsh and insists he was impressed with his side’s reaction.
“You can debate whether it was a yellow or not," Koeman said at full time. "It was a tough punishment, it’s difficult being a referee.”
"It's a decision we don't have to discuss.
"It was more important to talk about how to play with a man down than to discuss that.
"I'm very proud of this team in today's match.
"We worked very well and played with a lot of discipline in the team, with one man down.
"We won even with one player less.
"We're doing a job good.
"These results invite us to think that we're on the right track."
Barcelona have been drawn with Juventus, Dynamo Kyiv and Hungary side Ferencvaros in the Champions League group stage and Koeman was given the chance to react to the draw.
"We know that Juventus are one of the stronger teams in the group," he added.
"Let's hope we get through [to the knockout stages] in this competition, we have to get through."
