Reds' top scorer Nichols gave the visitors the lead on the 58th minute with an instinctive finish on the spin, after being found by George Francomb's cross, to finish off a slick team move.
Tranmere, who sit in fifth place in League Two, had ten shots in the game — two fewer than their opponents — but they were unable to find a way back to claim a fourth straight win.
It was a return to winning ways for Crawley who suffered a 3-1 defeat at Morecambe on Tuesday, and also marked their first league double of the season, having beaten Tranmere 4-0 at home in October.
The result leaves John Yems' side in 12th, seven points short of the play-off places.