Accies led at Gayfield in the 24th minute through Daniel O'Reilly but the hosts hit back from the penalty spot on 37 minutes after the referee spotted a handball in the box and Nicky Low converted from 12 yards.
The visitors stunned the leaders again just seconds before the break as Andy Ryan converted Jamie Hamilton's cross.
Arbroath piled on the pressure in the second half and it finally paid off in the 91st minute when a Brian Easton foul on Jack Hamilton earned a spot kick, which McKenna converted to secure a vital point in the race for the title.
Particularly as second-placed Kilmarnock failed to capitalise, Derek McInnes’ side crashing to a 2-1 defeat at home to Ayr United in the Ayrshire derby.
The Honest Men came from behind to claim a 2-1 win with Fraser Mullen's seventh minute opener cancelled out by James Maxwell before Patrick Reading struck the winner 10 minutes from time.
A Brian Graham hat-trick looked to have earned Partick Thistle victory at Inverness until an equaliser from Kirk Broadfoot deep into injury-time secured a 3-3 draw for Inverness Caledonian Thistle.
Graham twice put Partick ahead in the first half, only for Billy Mackay and and Shane Sutherland to hit back before the break.
Graham secured the match ball when he converted a 72nd minute penalty but Broadfoot rescued the hosts in the 94th minute, who stay third but are now without a win in eight.