Terry Butcher's Hibees looked on course to chalk up their fourth consecutive league win thanks to midfielder Liam Craig's double either side of half-time.
But Jackie McNamara's battling Tangerines side refused to give up and earned a point after David Goodwillie's 83rd-minute strike spurred them on to get an equaliser through Graham.
Everton boss Roberto Martinez was in the stand to run the rule over Dundee United's promising youngsters who had lit up the first half of the SPFL season.
United were aiming to avoid a fourth straight league defeat after going down to St Mirren, St Johnstone and Aberdeen over a disappointing festive period.
But they faced a revived Hibs side who arrived on Tayside still buoyed by their late 2-1 win over Edinburgh rivals Hearts at Easter Road a few days before.
It was a slow start from both teams and United's first hint of an opening came after 11 minutes, but Hibs defender Paul Hanlon intercepted Nadir Ciftci's attempted pass to Ryan Gauld and was able to clear the danger.
Hibs began to find their feet after that and came close to making the breakthrough in the 15th minute.
Craig's corner was met by James Collins and his downward header looked to be going in before it was hooked off the line by United midfielder John Rankin.
Midway through the first half, Gauld went on a darting run before Hibees defender Michael Nelson halted him in his tracks.
From the resultant free-kick, Gauld rolled the ball to Ciftci who tried his luck from 25 yards but Ben Williams held the ball comfortably in the visitors goal.
At the other end, Hibs won a corner which Craig swung over and home keeper Radoslaw Cierzniak came racing off his line.
Michael Nelson nodded towards goal but Cierzniak got back enough to turn the ball over the bar at the expense of another corner.
There was controversy on the half-hour mark as United pushed for the opener.
Ciftci struck a long ball forward for Gary Mackay-Steven which prompted Williams to run off his line before mis-kicking his attempted clearance.
Williams then grabbed Mackay-Steven who was about to home in on goal from just outside the box but the linesman raised his flag for offside and the visitors breathed a sigh of relief.
However, Hibs went on to take the lead through Craig in the 37th minute.
John Souttar attempted to play a one-two out of defence with Mark Wilson but the ball fell to Paul Heffernan who wasted no time in threading a pass through to Craig.
The midfielder then ran on to fire a 12-yard shot low into Cierzniak's far corner.
United suffered another setback a few minutes later when Ciftci was forced to go off inured and he was replaced by Graham.
Gauld had a half-chance to level soon after half-time but saw his 25-yard shot go past Williams' left-hand post.
It was end-to-end stuff by this stage and, in the 57th minute, Heffernan was next to try his luck and he let fly with a left-foot effort which skidded wide.
United made their second substitution on the hour when Goodwillie came on for Gauld in a bid to force an equaliser.
But it was the visitors who increased their lead soon after from the penalty spot.
Souttar was judged to have shoved Collins as they contested a ball in the air and, after referee Steven McLean had no hesitation in pointing to the spot, up stepped Craig to make it 2-0.
United almost pulled one back in the 79th minute when Mackay-Steven crossed and Hanlon turned the ball towards his own goal, only for Williams to pull of a reflex save.
But seven minutes from time Goodwillie gave United some hope when he latched onto Graham's cross before turning and firing past Williams from close range.
Then, in the 90th minute, United levelled when Goodwillie's 12-yard shot was turned in by Graham as the points were shared.