Burke stole a lazy pass near midfield and then took a few dribbles before blasting a roller from outside the area just inside the far post to give Philadelphia (4-3-1) a 1-0 lead in the 14th minute.
Monteiro, a 25-year old on loan from FC Metz through June, converted from the spot in the 45th for his first MLS goal after Jack Elliott drew a foul conceded by Zakaria Diallo.
Alejandro Bedoya tapped in an empty-netter to cap the scoring 57th minute.
Montreal (3-3-2) had its three-game unbeaten streak snapped and conceded a goal for the first time since a 7-1 loss to Sporting Kansas City on March 30.
Philadelphia's Kai Wagner was shown a straight red card, after video review, in the 90th minute.