The 4-2 victory was the Light Blues' sixth win on the bounce but it was far from an easy 90 minutes for the visitors who had to work exceptionally hard throughout on the Gable Endies’ 3G surface.
Lloyd Young's flicked finish on 15 minutes gave Montrose a shock lead but Lee McCulloch equalised from the spot to make the score level going in at the break.
Kevin Kyle and Dean Shiels then netted after the hour mark but a terrific goal from substitute David Gray made it a nervy last 15 minutes for the travelling Bears who had their hearts in their mouths when Garry Wood struck the crossbar before number 12 Scott Johnston hit the post deep into injury time.
At the death Gers youngster Robbie Crawford, who came off the bench, then fired home a fine fourth to complete the scoring.
The first half of this match was action packed with both sides creating a remarkable number of chances.
After just 60 seconds Wood's ambitious shot from wide on the right had to be watched by Neil Alexander as it landed on the roof of his net.
And in the very next attack Jamie Winter's powerful drive from 25 yards flew inches past the post.
Rangers responded and on six minutes Shiels also tried his luck from distance before Argyriou tested keeper Sandy Wood with another effort after an excellent flick on from Kyle.
A free-kick from Lee McCulloch then went high and wide before Montrose scored with 15 minutes on the clock.
A long throw-in from right back Stephen McNally wasn't dealt with by the Gers defenders and Young nipped in to flick the ball past Alexander from six yards.
The action at this stage was relentless and just two minutes after going behind Rangers should have been level when Little burst through on goal but his shot was brilliant blocked by Montrose’s impressive number one.
On 19 minutes the keeper was at it again, this time tipping a Macleod free kick onto the post after everyone in the 18 yard box had failed to get their heads on to it.
Two minutes later, however, and Rangers were back on level terms.
McNally’s clumsy challenge on Templeton resulted in referee Bobby Madden pointing to the spot and McCulloch took his penalty with aplomb, sending Wood the wrong way.
On 25 minutes Lewis Macleod had a great chance to put his side in front when he latched on to a through ball but again Wood was there to deny the youngster his second goal of the campaign.
Before the half hour mark a Templeton shot flew over the bar before Young brought out a terrific save from Alexander as this pulsating game flowed from end to end.
Into the second period and Montrose continued to trouble the Gers defence and on 49 minutes they should have got their noses in front.
A deep corner found McNally but the defender somehow missed the target from four yards at the back post with his downward header.
On 62 minutes McCoist decided to make his first switch with McKay replacing Templeton and his introduction proved pivotal.
Just four minutes later Rangers made it 2-1 when Shiels weaved through Montrose's defence and when the ball broke to Kyle his accurate low shot beat Wood to the keeper's left.
The goal was just what the Light Blues needed and two minutes later they had bagged a third.
McKay picked the ball up in his own half and after running 40 yards he played in Sheils who fired it into the opposite bottom corner of the net.
At this stage the game looked to be over but Montrose showed great fighting spirit and they equalised when substitute Gray hit a brilliant first-time shot which curled over Alexander into the top corner.
This only added to the tension in the latter stages and Rangers can count themselves lucky they didn't concede again when Wood's shot hit the cross bar before Johnston smacked the post in injury time.
In what was a dramatic finale to an entertaining contest Crawford then added a fourth which secured a big victory at the start of a busy period of festive fixtures.