The Hoops took an early lead through Gary Hooper but went in level at half-time after Ari deservedly levelled.
However, Celtic were much-improved after the break and Kris Common's 82nd-minute penalty saw them clinch their place in the knockout stages of the Champions League.
"It's fantastic," said Brown. "The lads worked so hard. In the last five minutes we thought the game was never going to end.
"Credit where credit's due, we worked really hard, created chances and scored goals. We knew we were good at home. The fans were phenomenal, they kept us going for the 90 minutes."
With Benfica drawing in Barcelona, Celtic pushed hard for a late winner and were handed a golden chance when Giorgios Samaras was felled in the box.
Commons' penalty smashed off the crossbar and into the net, much to the relief of the Celtic players and supporters.
"Once it hit the bar I thought it was going to bounce out, but I had faith in Chris," said Brown. "It was a great penalty."
Brown was forced off with six minutes remaining but he doesn't believe it will be rule him out of Saturday's trip to Kilmarnock.
"It was just cramp due to a lack of games," he explained. "I'm feeling miles better.
"I've been getting a few treatments here and there. I'm in good shape."