After falling behind to Ian Craney's 51st minute free-kick, Barnet hit back thanks to a magnificent 20-yard volley from Liam Hatch and Sagi Burton's close-range finish.
But their failure to put the game beyond doubt allowed Stanley a lifeline - and once Josh Wright and Jason Puncheon had both missed glorious chances - Roscoe D'Sane made the most of a goalkeeping error by Lee Harrison to scramble an equaliser.
Barnet should have taken the lead within the first three minutes of the contest after a dreadful piece of Stanley defending afforded Adam Birchall a clear run at goal.
A poor back-pass from Andy Proctor gave the Barnet striker the freedom of the Stanley penalty area - but the former Arsenal man could only fire straight at Ian Dunbavin from close range.
The hosts continued to press and Josh Wright saw his thunderous right-footed effort well saved by Dunbavin - before Joe Devera's mazy run, which saw him go past three red shirts, ended with the full-back dragging his effort wide of the far post.
It took nearly 20 minutes for the visitors to create anything of note - but D'Sane should surely have done better after being put clean through by Jay Harris' delightful pass.
The former AFC Wimbledon man sprinted clear of the home defence only for Harrison to produce a stunning save.
With half-time on the horizon, Stanley burst into life and should have gone into the break with a two-goal advantage.
John Mullin's impudent right-foot flick found Craney in acres of space inside the Barnet penalty area - but the midfielder somehow contrived to miss from six yards as he prodded his effort past the post.
If that was a let-off for Barnet, then what followed just seconds later was a get-out-of-jail free card - D'Sane burst clear through once again only to send a rasping effort against the outside of the post.
With Stanley still shaking their heads in disbelief, Liam Hatch came within a whisker of increasing their misery - but the Barnet striker fired his effort wide of the far post after finding himself clear on goal.
Barnet started the second half on the front foot - Puncheon's cross-cum-shot was parried by Dunbavin - only for the offside flag to go up after Hatch had fired home the rebound.
And seconds later Accrington found themselves ahead as Craney made up for his woeful miss in the first half with a wonderful curling effort from 20 yards.
Burton was adjudged to brought down D'Sane on the edge of the penalty area and Craney stood up to curl a delightful effort past the outstretched arm of Harrison.
Barnet should have claimed an equaliser just minutes later - Hatch saw his close-range effort tipped away by Dunbavin - only for Birchall to head the rebound against the post.
With Stanley collapsing against the ropes, it was only a matter of time before Barnet found an equaliser and when it came it was well worth the wait.
A cross from the left picked out Hatch and, after taking a touch to set himself up for the volley, the striker produced a magnificent shot which gave Dunbavin no chance.
And just moments later Barnet were celebrating again - Burton firing home an unstoppable effort from close range after an Ashley Carew corner caused all sorts of problems inside the Stanley penalty area.
Both Wright and Puncheon missed gilt-edged chances as Barnet threatened to run riot. But with time ticking down, the hosts were punished for their lack of a cutting edge - D'Sane scrambling home an equaliser to deny Barnet all three points.