Goals from Kris Commons and substitute James Forrest handed Ronny Deila his first trophy as Celtic boss and kept them on course for a treble.
But Johansen hailed Brown, who was pictured out drinking the week of the Cup final, saying what a terrific captain he is.
“I think Broonie showed what a captain he is today,” he said. “I thought he was absolutely fantastic.
“He’s 100 per cent in training every day so when that came up I wasn’t concerned, not even a little bit.
“It’s not what you want to see happening but it’s happened and you have to move on.
“But what he did today was unbelievable. He’s a captain with big letters. He showed today what a player he is.”
It was second meeting with Dundee United of four in a fortnight, with the Scottish Cup quarter-final replay to be played on Wednesday before the Premiership encounter next weekend.