John Baird gave the visitors - now 11 points adrift at the foot of the table - an eighth-minute lead at Easter Road.
However Griffiths netted four minutes into the second half to earn Pat Fenlon's men a point.
Barry Smith's Dundee side stay bottom of the league table, a long way behind 11th-placed Ross County still, but they can be encouraged by a performance that could have deservedly earned them all three points.
The first goal was scored before any rhythm could be established but the advantage the visitors held at half-time was more than deserved, Dundee playing with confidence and composure which their hosts could not match.
Former Hibs striker Colin Nish was the goal creator.
Much maligned in his time at Easter Road, Nish played the ball through to Steven Milne in a central area before drifting past a daydreaming crowd of opposing players, controlling the return ball and weighting a perfect through-ball for Baird who coolly slotted past goalkeeper Ben Williams.
A stunned home side should have equalised 60 seconds later. Tim Clancy seemed to repel potential markers as he drifted into the six-yard box but he headed David Wotherspoon's cross awkwardly over the bar.
At the other end Milne lacked the composure Baird had shown minutes earlier when he was presented with an almost identical chance. This time Williams was given a chance and impressively parried the striker's attempt away.
Jim McAlister was also found lacking five minutes before the interval. The midfielder blazed over after another shoddy example of defending allowed Nish to create a chance through a routine long ball from goalkeeper Rab Douglas.
One of the most startling aspects of Hibernian's play was provided by striker Griffiths, who posted a very passive performance during the opening period.
The Dundee defence had done an excellent job of turning him into potential creator instead of goalscorer.
However, conceding a free-kick 20 yards from goal was all the invitation Griffiths needed to level the scores. The Scotland international clipped the ball over the wall and past Douglas at his near post.
Dundee created the better chances thereafter, with Milne firing at Williams from close range and Nish seeing a shot parried by the Hibernian goalkeeper.