Mats Zuccarello had scored in the third period to level it for the Rangers, who started the season with nine straight away games because of renovation work being done at Madison Square Garden, before Brassard broke away to score with 12 seconds of OT remaining to seal a 3-2 victory.
Benoit Pouliot had a goal and an assist for New York, while Drew Miller and Daniel Alfredsson tallied for the Red Wings.
Logan Couture scored both goals as the San Jose Sharks saw off the Montreal Canadiens 2-0.
Patrick Marleau had two assists, while Antti Niemi made 22 saves for his second second shutout in three games.
Vinny Lecavalier scored a hat-trick as the Philadelphia Flyers broke out for a 5-2 win over the New York Islanders.
Jakub Voracek and Matt Read also tallied for Flyers as they scored more than two goals for the first time this season.
Andy Greene scored with 45 seconds remaining as the New Jersey Devils ended a seven-game losing streak by rallying from 3-1 down to beat the Boston Bruins 4-3.
Jason Pominville scored twice and Niklas Backstrom made 33 saves to get his first win of the season as the Minnesota Wild beat the Chicago Blackhawks 5-3.
Andrew Ladd scored the only goal in the shootout as the Winnipeg Jets beat the Dallas Stars 2-1.
Mike Cammalleri's double helped the Calgary Flames beat the Washington Capitals 5-2, while the Tampa Bay Lightning rallied to edge out the Buffalo Sabres 3-2.
There were also wins for the St Louis Blues, the Phoenix Coyotes and the Toronto Maple Leafs.