The coach, who took over in the wake of Neil Harris' resignation on Thursday, dedicated the win to the former manager.
READ: What Marcelo Bielsa said about today's defeat for Leeds"It's been a real topsy turvy and a bit of a whirlwind 24, 48 hours," he said.
"I wanted to put a performance on and the lads did for Neil and David [Livermore, Harris' assistant].
"They've been instrumental in this club, turned it around, but an infrastructure in place.
"I was proud we got a performance for them.
"They brought me to the football club, gave me an opportunity when I retired from playing. To get a victory for him meant an awful lot to everyone at the football club."
Barrett said the emotional circumstances surrounding the game had taken a toll, so he was proud of the effort Millwall put in against promotion hopefuls Leeds.
"They're a top side but this is always a game, the atmosphere is electric," he said.
"It was an emotionally draining 48 hours. A lot has gone on. We had to prepare the boys after the shock news.
"The character shown was outstanding. Maybe it was a good game because it was Leeds and there's that extra spice but take nothing away from the lads."
The game's talking point came on 14 minutes when Berardi lost the ball to Tom Bradshaw who went in on goal and then hit the deck.
Referee Linington pointed to the spot and produced a red card for Berardi.
Barrett agreed with the officiating.
"I thought it was a penalty first of all," he said.
"By letter of the law that is what it is. The ref has given it and you get on with it."
Millwall took the lead through Jed Wallace's spot-kick and went 2-0 up through a fine Bradshaw finish in first half stoppage time.
Leeds came roaring back, scoring within seconds of the restart through Gjanni Alioski.
The Whites dominated second half possession without ever really hurting the hosts, who managed the game well in the final stages, keeping the ball in the corner and preventing Leeds from playing out.
"The first half was tremendous, the intensity was first class, we didn't want to give them any breathing space," said Barrett.
"I thought we deserved to come in leading. I was delighted with the first half.
"I was trying to calm everyone down as they came in. The atmosphere was rocking.
"Everything we said didn't happen when we came out.
"We wanted a fast start, Leeds broke and scored so it was a terrible start to the second half and you have to regroup. I was really proud of the boys.
"Leeds took the handbrake off and played with freedom but the shape was excellent. We've limited them to hardly any attempts.
"The lads were tiring, the mental tiredness over the last couple of days.
"They dug in and grafted and we came strong at the end again.
"A little bit more composure would have killed the game off."