Lee Johnson’s second match in charge of the Easter Road club saw his players booed off the pitch by the away support, as they failed to break down a resolute Bairns backline.
The result, which marked McGlynn’s first competitive home match at the helm, sees Falkirk now sit on five points after two matches played in the group.
The home side pinned the Hibees back early on, with Callumn Morrison fizzing an effort just wide on 7 minutes.
Moments later, a mix-up between Kevin Dabrowski and Kyle McClelland gave Aidan Nesbitt a free shot at an open goal.
However, the midfielder somehow blasted wide of target with the goal gaping.
Craig McGuffie then tested the Polish stopper with an effort from range as Falkirk showed their attacking intent under McGlynn.
With just 20 minutes on the clock, the top-flight outfit were rocking.
McClelland didn’t look comfortable at the back and a clearance from young defender forced his own ‘keeper into a strong reflex save.
Hibs first real attempt came ten minutes from the break, with a tame header from Christian Doidge forcing a corner kick which came to nothing.
Then, a darting run from Leon McCann, who was the only chance from Saturday’s side, saw the full-back float an inch-perfect cross into the box.
An on-rushing Nesbitt was the unlikely man in the right place at the right time as he flicked a header over Dabrowski into the back of the net.
For the first time in a long time, the Falkirk Stadium erupted as the Bairns took a 1-0 lead on 39 minutes.
The performance was epitomised by a strong Sean Mackie tackle, perfectly timed, that sent Hibees new signing Lewis Miller flying.
In the second half, Johnson rung the changes with the likes of Ewan Henderson, Elias Melkerson and Chris Cadden coming onto the pitch.
The latter shone and he first tested PJ Morrison with a strike on 51 minutes.
Finn Yeats then cleared one off the line to halt Henderson from scoring from a long range effort.
It looked certain Hibs would grab an equaliser as they pushed forward.
Morrison made a stunning save on 71 minutes to halt the ex-Celtic midfielder who struck this time in the box.
A stramash followed and Brad McKay, who was a colossus at the back, cleared the ball of the line with Doidge ready to pounce.
Then with three minutes left, Norwegian forward Melkerson thought he had scored when he saw a shot bounce back off the post.
In the final seconds of the game, in the fourth minute of injury time, Doidge was sent through one-on-one.
But it was to be Falkirk’s night, with the Welsh target man failing to find the goal Hibs were desperately looking for.
McKay once again was the saviour clearing off the line, nearly bursting the net in the process.
The victory sees the Bairns now travel to Bonnyrigg Rose Athletic next Wednesday night as group leaders.
Falkirk: PJ Morrison, Finn Yeats, Brad McKay, Sean Mackie, Leon McCann, Craig McGuffie, Steven Hetherington, Stephen McGinn, Callumn Morrison, Aidan Nesbitt, Gary Oliver.
Subs: Paddy Martin, Seb Ross, Jaime Wilson, Blair Sneddon, Kyle Connolly, Pearse Carroll.
Hibernian: Kevin Dabrowski, Lewis Miller, Kyle McClelland, Rocky Bushiri, Lewis Stevenson, Jake Doyle-Hayes, Josh Campbell, Joe Newell, Runar Hague, Aidan McGeady, Christian Doidge.
Subs: David Marshall, Ryan Porteous, Nohan Kenneh, Chris Cadden, Ewan Henderson, Elias Melkerson, Dylan Tait, Allan Delferriere, Jacob Blaney.
Referee: Kevin Clancy.