New York opened the scoring late in the first period when Jimmy Vesey fed Desharnais for his fourth goal of the year to put the Rangers ahead 17:35 into the game.
Detroit evened the score 7:46 into the second on a goal by Andreas Athanasiou, and it remained that way until early in the third period.
Lettieri, who was recalled Thursday from Hartford, fired a low wrist shot from a feed by Boo Nieves that beat Jimmy Howard 2:35 in the final period.
Detroit again evened the score 4:30 later on a goal by Henrik Zetterberg.
Brady Skjei nearly won the game when he was awarded a penalty shot in overtime, but could not solve Howard.
Nielsen beat Lundqvist in the first round of the shootout, while Mats Zuccarello, Mika Zibanejad (missed) and Kevin Shattenkirk were stopped.
The Rangers face the Buffalo Sabres Monday in the 2018 Winter Classic.
