A stunning free-kick from Callumn Morrison put the Bairns ahead on the ten minute mark, but a converted spot-kick from Lewis Turner earned the Rosey Posey a deserved point at New Dundas Park.
Goalkeeper PJ Morrison made a key save in the shoot-out, saving Hunter’s poor effort – which allowed Gary Oliver to score the winning kick.
The match started with John McGlynn’s side on top, and it was Morrison who was leading the away sides attacking threat.
On the tenth-minute, his first effort on goal opened the scoring.
His dead-centre free-kick arrowed in off the post to beat a helpless Michael Andrews in goal.
Moments later, the former Hearts winger tested the stopper again with a decent effort from range.
Ryan Williamson’s smart one-two then saw Morrison fire straight at Andrews again as Falkirk looked for a second.
However, the Lowland League champions grew into the match as it went on and they began to cause the backline problems.
On 20 minutes, they though they had a leveller when Dean Brett’s cross sailed into the back of the net, but the goal was chopped off for a foul in the box.
The Bairns injury woes then added up once again, with Liam Henderson being forced off in his first match back since serving a suspension.
Youth prospect Blair Sneddon came into the pitch, leaving Falkirk’s back four with three left-backs in the defence.
Gary Oliver and Aidan Nesbitt both had shots at goal from distance as the first-half petered out, with the away side looking their grip on the game.
Then just before the interval, Robbie Horn's side were given a chance to equalise.
Captain Jonny Stewart’s attempted cross was handled in the box by Nesbitt, who could have no complaints.
Turner stepped up and slotted home the penalty to make it 1-1.
In the second half, the match was turned ugly by the League 2 side, who forced Falkirk back with their direct play, which was causing problems for a makeshift backline.
On the counter-attack the Bairns still looked dangerous.
Morrison should have put his side ahead again on 56 minutes when he cut inside and went one-on-one with Andrews.
However, his effort was weak and an easy save.
He had another chance again with fifteen minutes to go when substitute Jaime Wilson fed the ball through to the goalscorer on the night.
This time a much more powerful strike cause the stopper problems, but he managed to divert the shot out for a corner.
The hosts had some chances too. Ross Gray should have scored when he was given the chance to shoot unopposed in the Falkirk box but he fired his effort over the bar.
In the final seconds in injury-time, Connor Doan was denied a late winner by Morrison – who dived well to his left to send the match to spot-kicks.
Falkirk prevailed on penalties 4-2 to gain a bonus point, which now sees them able to win Group D with just a point on Saturday against Clyde.
Lee Johnson’s Hibs side fell to a penalty shoot-out defeat in their match against Greenock Morton.
Bonnyrigg Rose: Andrews; Young, Grigor, Doan; Brett, R Gray (Martyniuk 90), Stewart, Hall (Smith 66), Barrett; McGachie (Hunter 70), Turner (Connolly 70). Subs: Weir, Wilson, Hancock, Khan, S Gray, Smith.
Falkirk: Morrison; Mackie, McGinn, Henderson (Sneddon 22); Williamson, Yeats, Nesbitt (Wilson 74), Hetherington (McGuffie 63), McCann; Morrison, Oliver. Subs: Martin, Ross, Connolly, Carroll.