Nielsen preserved the point for Kansas City (16-7-8) in stoppage time when he stopped a header by Montreal's Matteo Ferrari off a free kick from Marco Di Vaio.
Kansas City extended its unbeaten streak to eight games and improved its lead in the Eastern Conference to five points with the victory, which was played in a soaking rain.
The expansion team, out of the playoff hunt in the East, went to 12-15-5 with the tie.
NBC reportedly holds celebs hostage to Jimmy Fallon's show
Millions of Getty images now available for free via embed tool