The Croatian left-back was shown to be crying, hugging members of the backroom team and crouched down on the turf before heading down the tunnel.
Barisic had provided the assist for Filip Helander to score the only goal of the game.
The player’s reaction has led to suggestions that the left-back could leave the club before the close of the transfer window on Tuesday night.
Speaking on Sky Sports, former Rangers striker Kris Boyd noted that the club may have to sell someone following their failure to reach the Champions League group stages.
"There's two minutes left of the transfer window,” he said. “If he's waving his goodbyes or saying cheerio to the Rangers fans... listen he's been a great servant.
"When they were knocked out of the Champions League, all the stories were that someone will have to go.”