Town fell behind midway through the first half and the home side doubled their lead from the penalty spot five minutes into the second half after Akinde was fouled in the area.
The Glovers pulled a goal back via a Matt Dolan 20-yard strike before Akinde made it 3-1. Yeovil then pulled another goal back thanks to Nathan Smith’s header befire pulling level with 15 minutes to go after Francois Zoko scored.
In the dying stages Harry Cornick made it 4-3 following a superb break from a Barnet corner.
Manager Darren Way made on change to his starting eleven with Marc Laird making a return in the place of Harry Cornick.
On five minutes the home side nearly opened the scoring when a corner from the left was met by the head of Dembele at the back post. His header looped over Krysiak but Ryan Dickson was on the line to head away any danger.
Barnet took the lead on 22 minutes after Town failed to clear their lines. A cross from the Barnet left from Akinde was met at the back post by James Pearson who crossed back across the goal into the path of Gash, whose shot hit the ground and looped over Krysiak into the top corner of the net.
Just after the half hour mark Town nearly drew level. A free kick on the right wing was swung in by Dolan and with Stephens back pedaling the ball bounced back off the cross bar and the home side managed to clear away to safety.
With five minutes of the first half remaining Town had their best chance of the half. A superb ball from Dolan found the run of Laird, but with just the keeper to beat the Scot sent his header over the bar.
Town started the second period very brightly and nearly equalised within seconds of the second half startings. Zoko beat his marker to the ball to get enough space in the area to fire a shot at goal. Keeper Stephens made a good save with his chest and the ball fell to Laird, but shot was well blocked by defender Nelson.
Despite the bright start by the Glovers to the second half the home side made it 2-0 after 52 minutes. Barnet striker Akinde was fouled inside the area by Alex Lacey leaving referee Stockbridge no option but to award a penalty. The former Bristol City striker dusted himself down to send Krysiak the wrong way to make it 2-0 to the home side.
Town pulled a goal back within five minutes when a 20-yard strike from midfielder Dolan beat the full stretched Stephens and nestled into the bottom corner.
Just afte the hour mark Akinde scored his second of the afternoon when he latched onto a long ball over the top to head over Krysiak into an empty net.
Moments later Town made it 3-2. A corner from the right was whipped in by goal scorer Dolan. Defender Smith timed his run right to meet the ball with his head to nod home past Stephens, to score only his second ever career goal.
Town made it 3-3 with 15 minutes remaining when Zoko twisted and turned his way into space to fire a low effort past Stephens, much to the delight of the noisy travelling fans in the away end.
The home side had the ball in the net in injury time, but scorer Akinde was deemed to have fouled Lacey in the build up.
Town secured the win in the dying stages when Krysiak gathered a corner and released Walsh, who broke quickly to release Cornick, who made no mistake in slotting home to make it 4-3.