However, the visitors put that disappointment behind to equalise 1-1 in the 25th minute through striker Tom Jackson, who latched on to a ball about 10m inside his half on the right flank before threading it between the legs of charging centreback Finlay Milne.
Jackson kept his line, skinned defensive midfielder Rob Pearson and left him and Milne clutching their hamstrings before crisply drilling the ball past goalkeeper Kyle Baxter.
Milne hobbled off as Blues assistant coach Chris Greatholder replaced him with James Hoyle but the Blues' defence not long after had started showing some cracks although they had dominated play in the first 15 minutes.
Minutes later Rangers had a casualty in Ben Thomas after fellow centre-mid George Milne clipped him between the eyes in trying to bring a high ball down but the former returned to play with a bandage around his forehead.
In the 36th minute Pearson, whose body language didn't look good after the equaliser, collected a yellow card for grabbing the shirt of a rampaging Bevin from behind.
Three minutes later, Bevin put a clever ball over his head inside the box but Dominic Rowe's ensuing shot went wide by a ball width as the Bluewater Stadium faithful breathed a sigh of relief.
English import Pearson ominously flirted with a red card after conceding another freekick for a silly foul.
Jackson's throws into the box, that would have put many cornerkicks to shame, alone were worth the fiver charged at the gate and something Rovers captain Danny Wilson and his troops had to keep in check to ensure their double hopes were kept alive.
In the 52nd minute, Blues striker picked up a yellow card for a sprigs-up tackle on defender Tom Probert.
Rangers captain Patrick Fleming put the visitors ahead, 2-1, in the 59th minute when substitute Richard Bott received an educated switch-play cross on the right flank following a Blues' blunder in their attacking third.
Bott comfortably beat defender Wilson into the 18m box from his flank, before deftly cutting the ball back to Fleming who slotted it past keeper Baxter who didn't move.
Miramar got their tails up and staged waves of attacks on the hosts' goalmouth as the Blues rode their luck.
In the 63rd minute, Baxter kept the Rovers in the game with a diving save to his right.
Substitute James Hoyle collected a yellow card in the 71st minute for a professional foul on an attacker who would have surged into the box on a counterattack had he not been tackled from behind.
In the 80th minute, the hosts had a great chance to equalise but striker Stephen Hoyle's effort after keeper Clarke fumbled the cornerkick went sailing above the crossbar.
But a minute later Tinsley equalised 2-2, to the delight of the parochial fans, from another penalty when Stoneman took exception to a sliding tackle from a visiting defender on the top of the 18m box.
Keeper Clarke this time picked the correct side to dive but the English import's shot was too good to the keeper's bottom left corner.
Not done, Bevin ran though in the 86th minute for a 3-2 lead when Pearson misheaded a long-range ball behind and over his fellow defenders.
Bevin casually took the ball around Baxter, who had slipped, to tuck it into the net.
But the home fans lifted their troops' spirits and two minutes later Tom Biss equalised 3-all, after a quick-flick header from Wilson, to jab the ball into the net at the far post.
In 30 minutes of extra time, Tinsley had a chance to keep the dream alive but his header on the ground bounced over the crossbar.
Teenager Ross Willox came in for Pearson in the rear of the engine room but Neil went down soon after to stop play not far from his box for some attendance.
A minute before the first half of extra time ended, Bevin put a hush through the crowd when he latched on to a long ball from Fleming to chip a golfing wedge-like shot over a charging Baxter in goal to seal the 4-3 victory.
Daniel Ball came on for Neil, going up front for midfielder Harry Morton who had pushed back.
The substitute striker had a great chance to equalise from a long ball but fell in the box as the Miramar bench missed a heartbeat before referee Stoneman carried on with the play with five left on the clock.
Ball picked up a yellow card for a crude challenge on George Milne close to the sideline as the Rangers sat back and gave two foragers the licence to employ delaying tactics at the top although time had rapidly become the Rovers' enemy.
The referee flashed a yellow card at keeper Clarke for stalling a goal kick before indicating two minutes were left.
It was over Rovers.
