The visitors went in front on 15 minutes when O'Neill collected Leroy Millar's headed knockdown before brilliantly turning away from Jimmy Callacher and then curling a measured shot beyond Nathan Gartside from inside the penalty area.
Larne threatened again when Matty Lusty got away from Kym Nelson and rolled the ball across the face of goal, but Carrick defender Callacher produced a timely sliding intervention.
However, the Larne strike duo weren't to be denied a second time and they combined again for Larne's second goal on 20 minutes. Leroy Millar slipped a pass into the path of Lusty, who burst into the penalty area before unselfishly rolling the ball in to O'Neill and he scored from close range.
Gartside made another crucial stop to block an O'Neill shot before Larne were awarded a penalty just after the half-hour mark when Ben Buchanan-Rolleston blocked Dylan Sloan's run into the box.
The game was delayed for six minutes while Carrick defender Mikey Place was stretchered off with an injury before Bent fired the spot kick into the top corner of the net.
The game looked beyond Carrick at that stage, but remarkably they pulled two goals back in first-half stoppage time.
Firstly, Rohan Ferguson dropped Aidan Steele's corner inside the six-yard area and Gibson was first to react to force the ball home from close range.
Just a couple of minutes later, Jordan Forsythe's long throw was flicked on by Seanan Clucas, once again Ferguson's handling wasn't secure and the ball was bundled home from a couple of yards out by Gibson before Bent could head the ball off the line.
In the second period, Sloan saw a shot tipped over by Gartside while at the other end, Kym Nelson's back post header from Steele's corner came back off the post before being scrambled clear.
Carrick continued to press for an equaliser with substitute Matthew Snoddy's shot blocked by the legs of Ferguson.
But it was Larne who got the decisive next goal on 69 minutes when O'Neill volleyed home after Ronan Doherty's long throw was flicked into his path.
Larne wrapped up victory with nine minutes left when Bent scored his second penalty of the afternoon after Crowe had been sent off for pulling back Sloan inside the penalty area.