The winner was set up by defenseman Michael Del Zotto, who held the puck near the blue line after a dish from Brandon Dubinsky up the left wing. He centered for Callahan, whose rising wrister hit the back of the net just under the crossbar.
It assured the Blueshirts their first win over their Original Six rivals since October 25, 2003, and capped the Rangers captain's 27th birthday.
Zetterberg's wrister from the inner edge of the right circle slipped between Lundqvist's pads at the 3:36 mark of the opening period.
Richards tied the game during a power play on an end-to-end rush with just over five minutes remaining, eventually moving past two defenders in the Detroit zone and sliding a shot through Conklin's pads.
Lundqvist stopped 15 shots over the next 40 minutes and was saved late in regulation by a shot clanging off the crossbar. Conklin was equal to 17 shots.
Conklin went low to stop Marian Gaborik off a 2-on-1 just over a minute into overtime, then dove to his left seconds later to deny him again.