The match was their first competitive outing since the club was consigned to liquidation over the summer and voted into the Third Division.
Andy Little celebrated his return to the club with the opener after just four minutes but Rangers were forced to negotiate the first round tie the hard way when Andy Jackson levelled for the Second Division hosts. It took 102 minutes of football before McCulloch found the winner to finally allow Rangers to begin a new era on a positive note.
New recruits Ian Black and Little were both named in the starting line-up for the Glasgow giants. Former Hearts midfielder Black joined the Ibrox side on Saturday and striker Little re-signed after his Light Blues contract expired at the end of last season.
Both featured as trialists as they cannot be registered until Friday when 'oldco' Rangers' SPL share is formally transferred to Dundee.
Defenders Carlos Bocanegra and Dorin Goian - both expected to leave after several first team regulars refused to transfer their contracts to the newco - also started at Glebe Park. There were other familiar faces in the starting XI - McCulloch, Lee Wallace and Kirk Broadfoot - as well as youngsters such as Barry McKay and Lewis MacLeod.
The breakthrough came after just four minutes when Little raced onto a McCulloch ball and kept his cool to slot past the Brechin number one.
However, if the away fans expected a flood of goals to follow they were disappointed, with Brechin hauling themselves level two minutes before the break. Ewan Moyes delivered the pass down the left flank and Goian's indecision allowed Jackson to unleash from a tight angle past the grasp of Alexander and into the back of the net.
With just over 10 minutes to go, Rangers fans claimed Paul McLean handled a McCulloch shot in the box but there was no penalty from referee George Salmond. With neither side able to claim a winner inside 90 minutes, the tie was forced to extra time. City looked lively and David McKenna's shot for the home side deflected just over the crossbar.
Instead, it was Ally McCoist's men who had the net bulging after 102 minutes when McCulloch rose above everyone else in the packed goalmouth to nod home from a MacLeod corner and finally grab a winner for Rangers.