Akinde had earlier seen his opener cancelled out by Joe Healy, but the in-form forward earned Martin Allen's team a dramatic win in stoppage time to maintain their nine-point advantage at the top of the table.
After already recording 12 wins from 18 league games, Barnet started confidently at Park View Road, with both Lee Cook and Akinde coming close to giving the Bees the early advantage.
Ben Jefford offered a brief response for Welling but, shortly after the midway point of the first half, Barnet edged ahead when Akinde powered the ball into the net from close range for his 10th strike of the campaign.
However, as the match approached the interval, Welling began to improve with Harry Beautyman having a volley saved by Graham Stack, but the goalkeeper was beaten a minute before the break when he fumbled a free kick to allow Healy to net the rebound.
Stephens wasted another headed opportunity for the visitors just after the hour mark, but Barnet were struggling to break down a resolute Welling side who were unbeaten in eight home fixtures.
As the game entered the closing stages, the pressure was coming from Welling and Kieron St Aimie wasted a glorious chance at the back post to nick a winner against his former side.
Welling were on the verge of extending their impressive streak at home, but a minute into added-on time, Akinde found the bottom corner from 12 yards after being set up by Charlie McDonald's headed pass.