New York defenseman Mike Sauer had trouble handling the puck deep in his own zone and Connolly was standing in front of the net to fire a shot past Henrik Lundqvist 5:12 into the middle frame, handing Toronto a 2-0 lead.
Lupul made it 3-0 only 1:37 later on a power-play tally for the Leafs, who snapped the Rangers' seven-game home winning streak.
Jonas Gustavsson turned aside 30-of-32 shots to earn the win.
Artem Anisimov and Marian Gaborik made it close with third-period goals and Lundqvist finished with 26 saves for the Rangers.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut