It did not take long for the Premier League newcomers to open the scoring, as Cornelius scored his first goal for the club in the 16th minute by heading home Kevin McNaughton's sumptuous cross.
The home side thought that they had equalised through Ben Wright as the game approached the half-hour mark, but the referee disallowed the goal after spotting a foul.
Five minutes later, Cornelius's strike partner Maynard poked home Kim Bo-Kyung's cross to double Cardiff's advantage.
However, James Norwood stretched out to poke the ball home and pull a goal back for the home side with half time approaching.
Cardiff manager Malky Mackay made 11 changes to his side at half time, and it took just seconds for them to extend their lead to two goals.
Joe Mason was brought down in the penalty area and Peter Whittingham stepped up to regain the two-goal advantage.
The Blue Square Bet Premier side once again reduced the deficit through Danny Wright, but Craig Noone's strike from 25 yards once more extended Cardiff's lead.
Rovers refused to be beaten, though, and Matty Taylor fired home the seventh goal of the evening to serve up a tense finale for the 1,358 spectators at The New Lawn.
Cardiff threatened on the break as the hosts probed for an equaliser, but the Bluebirds held on to secure an entertaining win.