The Jambos - who had won the last four meetings between the Edinburgh rivals - took the lead through Ryan Stevenson, before being reduced to 10 men when Marius Zaliukas saw red in Sunday's SPL clash.
Goals from Liam Miller - from the penalty spot - and Ricardo Vaz Te looked set to hand Hibs a long-awaited victory over their city rivals until Elliott struck late on.
Hearts made four changes from their win over St Mirren. Goalkeeper Marian Kello was out with a back problem and was replaced by Jamie MacDonald.
Andrew Driver, Ismael Bouzid and Stevenson also started, with Rudi Skacel injured, Adrian Mrowiec suspended and David Obua dropped to the bench.
Hibs added Martin Scott and Miller to the side who drew with St Johnstone in their last game on March 5.
The home side could have grabbed the advantage with just a minute gone when Derek Riordan found the goal at his mercy only to sidefoot wide of target when he should at least have tested MacDonald in the Hearts goal.
At the other end, Elliott pounced on a cut-back from Driver only to fire straight at Mark Brown, the Hibs goalkeeper doing well to block the point-blank effort.
Scott picked up the first booking with only eight minutes on the clock for a challenge on Ian Black who, after a few minutes of concern for Hearts, was able to continue.
As the match continued at a frantic pace, Elliott went to ground in the box under pressure from Paul Hanlon but there was no penalty award for the visitors.
It was only a matter of time before the deadlock was broken and it was Hearts who took the lead with 24 minutes gone.
Driver provided the cross and Elliott touched on for Stevenson to rifle home an impressive left-foot shot from the edge of the area.
But the Jambos' joy was short-lived when Zaliukas was shown a straight red card for a foul on Akpo Sodje in the box, as referee Iain Brines pointed to the spot with 33 minutes gone.
Miller stepped up for the penalty and, despite a delay as the match official insisted he reposition the ball on the spot, he eventually lashed confidently high into the net to level the scoreline.
Hibs made a change at the interval when Scott was withdrawn for the introduction of Vaz Te, before they squandered a great chance to take the lead.
Goalscorer Miller turned provider when he slipped the ball into the path of Victor Palsson who only had the goalkeeper to beat but he blasted over.
Hibs continued to press and Vaz Te saw his volley blocked, before Palsson was replaced by David Wotherspoon for the final 20 minutes.
Riordan could have been the hero for Hibs when he collected Vaz Te's pass only to see his shot blocked well by MacDonald, before rifling over the rebound.
The honours went to Vaz Te instead with 10 minutes to go after he ran onto a lofted pass from Wotherspoon and stroked home from six yards.
But Hearts had the final say four minutes later when Craig Thomson's free-kick was touched on by Webster for Elliott to volley over the line from close range to seal a point.