Strikes from Charlie Mulgrew, Joe Ledley, Amido Balde and an own goal from Lee Ashcroft saw Celtic extend their lead at the top of the Premiership to 18 points and continue their unbeaten league season.
And goalkeeper Fraser Forster has now kept nine consecutive clean sheets and 11 in domestic competitions.
Brown told Celtic TV: "It's a great achievement. Looking around our dressing room and seeing what it means to the players, it shows that we have so much to play for this season.
"It was a great result against Kilmarnock. The lads played well from start to finish and I think it was played at a good tempo as well. We were playing one and two-touch on the pitch and we looked really good.
"There was one chance for Kilmarnock at the back post but we can always rely on Fraser at times like that because he always produces the big saves at the big times.
"With this run of results and the added achievement of the clean sheets, I think this is one of the best Celtic teams in a long time, so hopefully we can keep going for as long as possible."