Aberdeen manager McInnes clashed with the Hearts assistant boss after a late goal by Jordan McGhee earned Hearts a 2-1 victory.
The pair had to be separated by police and referee Alan Muir sent both to the stands following the bust-up.
After the match McInnes apologised to the referee, but not to Brown, insisting: "I have never seen an opposition member of staff come into their rivals' technical area to celebrate like Brown did.
"But as soon as I've put my arms on him to push him back I'm guilty."
At the hearing at Hampden Park, both men admitted breaching Disciplinary Rule 203 relating to 'misconduct at a match by leaving the technical area in the absence of special circumstances and by adopting an aggressive attitude to a member of the opposing team staff'.
The suspended ban will only come into force if either breach the same rule before the end of the season.