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 yesterday 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 just before the interval.
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 yesterday 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.
Rangers began the match like a team simply desperate to play football again following months of turmoil off the park.
McKay shot high and wide, before Little forced an early save from goalkeeper Michael Andrews.
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, and the home side almost drew level when Jackson set up Craig Molloy for a low shot that was gathered by Neil Alexander.
Attention was briefly drawn away from the action on the park when a fan within the travelling support climbed part of the way up a floodlight pole in front of the ground's famous hedge before being summoned down again by awaiting police.
On the park, Broadfoot cut inside from the right flank and forced a decent save from Andrews and the keeper was up to the task again to then block a thunderous long range effort from Wallace.
Rangers were left rueing those missed opportunities when Brechin hauled 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.
After the restart, Black had a decent chance to open his goalscoring account with a free-kick just outside the box but his curling effort was tipped just wide.
The former Jambo then squandered a decent chance when he met a Wallace cut-back in front of goal but failed to keep the effort on target.
At the other end, a Jonathan Brown free-kick from 30 yards out was smothered low by Alexander at his right-hand post.
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.