Joe Mason proved a handful for the Derby defence in the first period, and he almost broke the deadlock in the 16th minute when he latched onto a pass from Lee Chung-Yong and forced Derby stopper Lee Grant into a save at his near post.
Martin made amends for his penalty miss seven minutes before the break when he gave the Rams the lead, turning in Cyrus Christie's cross from inside the six-yard box to leave the hosts with it all to do in the second half.
Lonergan came to Bolton's rescue again just after the hour mark when Jordon Ibe charged into the penalty area and let fly, forcing an impressive save from the Trotters goalkeeper, but he could do nothing about Derby's second.
Martin doubled the Rams' advantage in the 57th minute when he raced through on goal and got the better of Lonergan in a one-on-one situation, putting Steve McClaren's side firmly in the driver's seat.
Ibe had another crack at goal in the 74th minute, cutting in from the left before unleashing a blistering strike that needed a reaction save from Lonergan to keep out, but it would have had little bearing on the outcome of the match.
Neil Danns came close to pulling one back for Bolton with a bending free kick late on, but Lee Grant was equal to it and Derby came out worthy winners.