Frans Nielsen had a goal and an assist while Matt Moulson also lit the lamp for the Islanders, who have dropped their past four games and 10 of 11, overall. Evgeni Nabokov allowed three goals on 29 shots in the loss.
The Islanders had tied the game at 2-2 on the power play early in the third, but Richards put the Rangers up for good with 4:55 left in the game.
Brandon Dubinsky held the puck in the slot and sent a pass over to the left for Richards, who beat Nabokov cleanly with a slap shot for his sixth goal of the year.
The Islanders pulled Nabokov late in the game and had several opportunities, including a clear shot that Andy MacDonald rang off the right post and Callahan's empty-netter just crossed the goal line before time expired.
The first goal of the game went to the Rangers as Avery's initial shot from the left side bounced off of Nabokov's glove and came right back to Avery, who buried the next attempt at 14:43 of the first.
The Islanders, though, got the goal back just over two minutes later when Nielsen skated down to the inside right circle and snapped a shot over the glove of Lundqvist.
The Rangers made it a 2-1 game 6:07 into the second as Dubinsky had the puck along the left boards and was able to fight over his checker before sending it to the right where a cutting Eminger wristed it home.
The Islanders again tied it at the 6:58 mark of the third with a power-play goal. Nielsen held the puck at the right point and sent it down low for John Tavares, who flung it across the goalmouth for a redirect by Moulson.
It's the longest winning streak for the Rangers since ripping off seven straight from Oct. 3-17, 2009...The Rangers took four of six from the Isles last season, but dropped the first game of the season between these two on Oct. 15...The Islanders host the Canadiens on Thursday...The Rangers play in Montreal on Saturday.