The Parkhead men were losing 3-0 at the interval at Rugby Park and looking like they would go 13 points behind leaders Rangers, who were playing St Mirren at Ibrox later in the day.
But in a dramatic second-hand turnaround, Celtic scored three times, with Mulgrew levelling with a close-range header, while St Mirren grabbed a late equaliser in Govan to keep the Hoops 10 points adrift of leaders Rangers and three behind second-placed Motherwell, although Celtic have a game in hand over both.
The Parkhead club's stand-in skipper is looking to replicate the second-half performance in Ayrshire against the French side, who are above bottom side Celtic in the section only on goal difference.
"Due to the spirit we showed in the second-half against Kilmarnock, we are very positive and looking forward to tomorrow's game," he said.
"We are glad to be in Europe, it is a big game and very important.
"A win would be massive for us and we will be looking to win.
"I'm sure it will be a very difficult game but we will be doing our best to win and it will be a great boost to our season if we could do that."
Mulgrew insists the changing fortunes of the Old Firm on Saturday gives him belief that Celtic are far from out of the Scottish Premier League title race.
"It just shows everybody that anything can happen," said the former Aberdeen defender.
"Down to the last kick of the ball, anything can happen.
"There is a long way to go in the season and we have to keep that in our minds and be focused for 90 minutes in every game."