The ninth minute saw the Grecians take the lead as Danny Butterfield found Nichols, who turned on-loan Leicester City defender Alie Sesay before smashing home the opener.
Cambridge defender George Taft saw yellow for a 15th-minute challenge but Nichols could not double his and Exeter's tally from the resulting free kick.
Barry Corr had the opportunity to level the scores five minutes later but failed to find the target from close range.
The second half saw Cambridge go on the offensive, as Exeter kept it tight in defence on the back of their one-goal lead.
Irishman Corr was on the end of another Cambridge attack but could only head straight at Olejnik.
In the 80th minute, the striker then headed down a ball into the path of Jordan Slew, who fired his effort over the bar.