The visitors broke the deadlock after 62 minutes as Dennis Appiah picked out Pieter Gerkens with a cross in the centre of the box and he headed home — with a touch off Hoops defender Jozo Simunovic — to hand the visitors the lead.
Anderlecht captain Sofiane Hanni tested goalkeeper Craig Gordon as the half wore on, moments after a chance at the other end from Mikael Lustig but it was to end just a one-goal match at Parkhead.
Brown said: “We knew what we had to do and had to dig deep at some point and we managed to get through to the Europa League and that was our aim at the start.”
Brown, when asked if it has been a bit of a frustrating night, said: “Yes, a little bit. The first half especially, we didn’t manage to keep the ball.
“They played their formation and kept the ball really well and were very, very patient and looked like they were the team that was going to score a goal in the first half. We still tried to play our normal game and play out from the back. They pressed us really high up the park and put us under a lot of pressure.”
Goalkeeper Gordon was delighted to keep his team in the game with an early fine save from midfielder Sven Kums, who blasted in a cut-back from Henry Onyekuru.
He said: “It was one of my first touches in the game. It was a great start from them in the game where we didn’t get many touches of the ball.
“They found themselves a big chance against us inside the first couple of minutes and I’ve thankfully come up with a big save.”
“Yeah, we didn’t get out from the back as well as what we normally do. That’s disappointing.
“We probably did it better in the second half at times.
“Sometimes, in a game like this, when there’s not a lot to lose and maybe people don’t want to make mistakes, it’s a bit more difficult to take the ball under pressure - but we’ve got to learn from that and be able to do it.”
Celtic manager Brendan Rodgers added: “Overall, the best team won, but the bigger picture was to get through to the Europa League by finishing third and that would be a massive achievement.
“We were the fourth seeds coming into this and there is immense pride that we got the job done.
“But we have a lot of work to do at this level, but we can look forward to the Europa League in February.”
Gordon, Lustig, Boyata, Simunovic, Tierney, Forrest, Brown, Armstrong (Ntcham 45), Sinclair (Rogic 45), McGregor, Dembele (Edouard 74).
Boeckx, Appiah (Bruno 90), Mbodji (Josue Sa 75), Spajic, Obradovic, Gerkens (Teodorczyk 78), Dendoncker, Kums, Trebel, Onyekuru, Hanni.
Matej Jug (Slovenia).