The Bhoys beat Ajax at Celtic Park with James Forrest's penalty and Beram Kayal's strike meaning that Lasse Schone's strike mattered little.
The win gives Celtic a foothold in the group after losses to AC Milan and Barcelona and will gave them something to build on when they head to Amsterdam next.
"It's always good to get a goal before half-time and there was pressure on him (Forrest)," Mulgrew told Sky Sports of the opener.
"He put it away brilliantly.
"We knew this would be a massive game. We needed the points, got them and got two good goals."
Forrest had a long wait before he could take the penalty and was happy to hold his nerve.
"I just focused on where I could put the ball and it landed in the back of the net," he said.
"It was brilliant to get a goal and we look forward to the next game away to them."