Scotland international Brown has faced Pirlo on international duty and for Celtic against AC Milan, and will meet again when Juve visit Parkhead in the first leg of the last 16 on February 12.
Last season, the 33-year-old inspired the Turin side to their first title since their demotion following a match-fixing scandal, as they went through the Serie A campaign unbeaten.
Pirlo then shone at Euro 2012 as Italy reached the final in impressive fashion before coming unstuck against Spain.
Speaking at Celtic's Lennoxtown training complex after the draw, Brown said: "In my position he's probably the one that I would say I would most like to be, but it's going to be me against him and I want to be coming out the winner at the end of that.
"It's been good playing against him and I learned a lot.
"He's their main player week in week out. He's the one that makes them tick. He's the one that threads the through balls and gets on the ball to start off the moves.
"He's the one that we are going to really have to watch out for."
Brown and Georgios Samaras are the only members of the Celtic squad to have played when the club last reached the knockout stages, before losing to Barcelona five seasons ago.
And the 27-year-old believes his current team-mates, having beaten Barca in the group stages, can go one better.
"It's huge," he said. "Some people never make this. I have been lucky enough to do it twice. The first time was great, we played Barcelona and were unlucky.
"Now I think this team is better than what we had before.
"We know it is going to be hard, but we have played really well at Celtic Park and hopefully teams fear coming to Celtic Park now.
"We believe in ourselves. We've got great legs all over the park, we can press teams high up the park and we can also get back.
"We always believe, we have faith in ourselves and as individuals as well.
"If you see our team spirit, we are out in a nice, rainy day in Lennoxtown and the lads are still looking forward to training and enjoying every moment.
"Pretty much everyone in the dressing room seems to have improved as players throughout the campaign."