Billy Kee notched his 14th goal of the season to give Accrington the lead when he latched onto Brad Halliday's cross.
Matt Crooks doubled their lead from 12 yards before Akinde found Luke Gambin, who fired past keeper Ross Etheridge.
And Akinde rescued a draw for the Bees as he rose highest to head home Mark Randall's late free-kick.
Barnet manager Martin Allen told BBC Radio London:
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"What a comeback that was. 2-0 down. Brilliant. You've got to say it was a brilliant tactical change by the manager!
"It's got to be one of the highlights of my 500 games.
"To change it and play with three up from when we're 2-0 down with 10-men and get a draw, it's a great effort from all the players and all the staff."
Striker John Akinde scored a 90th-minute equaliser as Barnet came from behind to earn a point against League Two promotion-chasing Accrington.