Grimsby striker Bogle saw his curling, left-footed free-kick tipped over the crossbar by Jamie Stephens before he opened the scoring after he burst past two defenders and had his shot deflected off Elliot Johnson and into the net.
And he doubled the lead 10 minutes before the interval when a mistake by Harry Taylor allowed Bogle to run free and beat Stephens.
The Bees got themselves back into the game in the 50th minute when Akinde burst into the box and was brought down by Josh Gowling. Akinde picked himself up to dispatch the penalty past James McKeown.
Akinde then levelled from the spot five minutes later after Danny Collins fouled Curtis Weston.
Ryan Watson then squandered two chances to win it for Barnet before the late drama.
Barnet were reduced to 10 men in the 90th minute when Ali Sesay, who had only been on the pitch for five minutes, was shown a straight red card for a challenge on Scott Vernon.
Grimsby were then given a golden chance to win it two minutes into stoppage time when Michael Nelson brought down Gowling, but Bogle sent his spot-kick over the bar.
Report supplied by the Press Association.
Match ends, Grimsby Town 2, Barnet 2.
Second Half ends, Grimsby Town 2, Barnet 2.
Danny Collins (Grimsby Town) is shown the yellow card for a bad foul.
Foul by Danny Collins (Grimsby Town).
John Akinde (Barnet) wins a free kick in the defensive half.
Penalty missed! Bad penalty by Omar Bogle (Grimsby Town) left footed shot is just a bit too high. Omar Bogle should be disappointed.
Penalty Grimsby Town. Josh Gowling draws a foul in the penalty area.
Penalty conceded by Michael Nelson (Barnet) after a foul in the penalty area.
Alie Sesay (Barnet) is shown the red card.
Scott Vernon (Grimsby Town) wins a free kick on the right wing.
Foul by Alie Sesay (Barnet).
Attempt missed. Omar Bogle (Grimsby Town) right footed shot from outside the box is high and wide to the right.
Scott Vernon (Grimsby Town) wins a free kick on the right wing.
Foul by Bira Dembélé (Barnet).
Foul by Omar Bogle (Grimsby Town).
Alie Sesay (Barnet) wins a free kick in the defensive half.
Substitution, Barnet. Alie Sesay replaces Curtis Weston.
Attempt missed. Brandon Comley (Grimsby Town) right footed shot from outside the box is too high.
Zak Mills (Grimsby Town) wins a free kick in the attacking half.
Foul by John Akinde (Barnet).
Attempt missed. Craig Disley (Grimsby Town) header from the centre of the box misses to the right.
Corner,  Grimsby Town. Conceded by Mauro Vilhete.
Attempt saved. John Akinde (Barnet) header from the centre of the box is saved in the centre of the goal.
Tom Bolarinwa (Grimsby Town) is shown the yellow card.
Foul by Tom Bolarinwa (Grimsby Town).
Sam Muggleton (Barnet) wins a free kick on the right wing.
Substitution, Grimsby Town. Scott Vernon replaces Kayden Jackson.
Attempt missed. Zak Mills (Grimsby Town) left footed shot from the centre of the box is close, but misses to the right.
Attempt missed. Jean-Louis Akpa-Akpro (Barnet) right footed shot from the centre of the box is close, but misses to the left.
Attempt missed. Zak Mills (Grimsby Town) right footed shot from outside the box is close, but misses to the right.
Tom Champion (Barnet) is shown the yellow card.
Omar Bogle (Grimsby Town) wins a free kick in the defensive half.
Foul by Mauro Vilhete (Barnet).
Zak Mills (Grimsby Town) wins a free kick on the left wing.
Foul by Jamal Campbell-Ryce (Barnet).
Attempt missed. Ryan Watson (Barnet) right footed shot from outside the box is too high.
Brandon Comley (Grimsby Town) wins a free kick in the attacking half.
Foul by Tom Champion (Barnet).
Attempt missed. Ryan Watson (Barnet) right footed shot from the centre of the box is close, but misses the top left corner.
Attempt saved. Curtis Weston (Barnet) right footed shot from outside the box is saved in the centre of the goal.
Omar Bogle scored twice before missing an injury-time penalty against 10-man Barnet, as Marcus Bignot picked up his first point as Grimsby Town manager.