Nicholas was 3 of 8 from deep while Steele was 4 of 9.
Lafayette (1-6) led 26-24 at halftime but Monmouth opened the second half with an 18-9 run to go up 42-35 and pull away with 12:58 left to play.
Stephen Spinella had six points during the run, Nicholson chipped in four and Steele capped the spurt with back-to-back 3-pointers.
Spinella, Steele, and Nicholson combined for seven of Monmouth's 19 steals while Dion Nesmith led the way with six steals of his own.
Monmouth (3-4) has won three straight games after a 1-4 start.
Tony Johnson led Lafayette with 14 points, nine rebounds, seven assists and five steals. Johnson is shooting 56 percent (34 of 60) from the field this season.
Authorities say former Michigan State basketball player Keith Appling has been charged with carrying a concealed weapon and other crimes.
Point guard Jaylen Fisher, who reopened his recruitment after coaching changes at UNLV, has committed to TCU over a long list of schools.
Forward Kennedy Meeks, who has averaged 9.4 points and 6.5 rebounds at North Carolina, has pulled out of the NBA draft and will return to the Tar Heels.