Stalock stopped Cizikas' backhander as the forward cut in off right wing. Cizikas didn't slow down, colliding hard with Stalock and propelling him into the right post.
Stalock remained sprawled on the ice for a couple of minutes, then skated to the bench. He was replaced by Arvid Soderblom. Cizikas was assessed a major penalty for goalie interference and a game misconduct.
The Blackhawks are already without top goaltender Petr Mrazek, who has a groin injury.
Across the ice, New York's Ilya Sorokin made 21 saves and lost a bid for his second shutout in three starts when Jonathan Toews scored a power-play goal on a deflection midway through the third period.
The Associated Press contributed to this report.