However, the 28-year-old will appeal the suspension, which would see him miss Tuesday's Group H home match against Ajax, the return in Amsterdam and the home tie with Milan on November 26.
Brown was sent off for kicking out at Neymar after the pair tangled while chasing a loose ball during the second half at Parkhead earlier this month.
The game was goalless at the time, and Barca went on to win 1-0 thanks to a late Cesc Fabregas header.
In a statement Celtic said: "Scott Brown has been given an additional two-game ban, on top of an automatic one-game ban, following his red card in the recent UEFA Champions League game against Barcelona at Celtic Park.
"Celtic can confirm that they have received notification from UEFA of the sanction. The club finds this to be excessive and they will be appealing against the decision."
Lennon critical of decision
Speaking immediately after the match on October 1, Lennon said: "I think the red card is very soft. If there was any contact, it was minimal. There was a lot more in a game that warranted heavier punishment.
"I don't think the initial tackle was a severe one. It is a physical game. I don't think Neymar does himself any favours with the way he behaves at times."
Meanwhile, Lennon has praised Brown and team-mate Charlie Mulgrew for the role they have played in Scotland's resurgence under manager Gordon Strachan.
Scotland's hopes of reaching the World Cup finals in Brazil were already over when Strachan took over in January but they ended their campaign with an impressive 2-0 win over Croatia on Tuesday.
Lennon said: "It was an excellent performance, excellent result and a great way to finish the campaign and Scotland should be very optimistic going forward.
"Charlie has been great. Talking about a multi-functional player, there is none better than Charlie.
"He can play centre-half, left-back, centre-midfield, wide left; he is a very intelligent player, technically brilliant and he has matured into this very good all-round footballer.
"So I am really pleased with him and obviously Scott captaining the team as well, he has consistency in his game at the minute.
"I thought Scott was the best midfielder on the park in Milan (in the Champions League) so those two are in a very good place."