Gage Ramsay, Josh Paterson, Mark Rubinchik and Braylon Shmyr scored for the Blades at SaskTel Centre.
Mark Kastelic got a shorthanded goal for Calgary, while Carson Focht tied the game up at 4-4 in the third period for the visitors.
Nielsen netted the game-winner 1:58 into the extra period.
Saskatoon goalie Nolan Maier stopped 30 shots in front of a home crowd of 3,579 Western Hockey League (WHL) fans. Nick Schneider made 20 saves for the Hitmen (21-36-10).
The Prince Albert Raiders (31-25-11) sit five points ahead of Saskatoon in the hunt for a WHL playoff spot.
The Blades (32-31-4) host the Brandon Wheat Kings (36-26-5) on Saturday.