Berra turned a Daryl Horgan cross into his own net just minutes after Marc McNulty had missed a penalty for Hibs but, just as it looked as if Paul Heckingbottom's side had clinched the points to close in on fourth-placed Kilmarnock, Ikpeazu slid home substitute Steven McLean's pass.
A first half which was littered by errors and misplaced passes from both sides finally saw two moments in which either side could have taken the lead. Hearts’ Jake Mulraney embarked on a run which took him past three opponents before he unleashed a shot which had goalkeeper Ofir Marciano leaping acrobatically to his left to push aside.
Few would have put Lewis Strevenson down for the opening goal, but the Hibs defender took a free-kick from Stevie Mallan, thundering in a shot from fully 30 yards which Hearts goalkeeper Zdenek Zlamal could only parry, the ball eventually falling for McNulty, who saw his bid to turn it back across goal blocked by Berra.
Up to then, any problems Craig Levein's players had endured had been largely of their own making, John Souttar's lack of control presenting Stephane Omeonga with the first opening of the game but the little midfielder was forced wide and Zlamal easily knocked his shot from a tight angle to safety.
A timely tackle by Paul Hanlon prevented Ikpeazu from capitalising on Connor Smith's ball before Daryl Horgan flicked Omeonga's pass over and then Berra came over to cover as Flo Kamberi threatened. Other than that, though, it was down to those final two minutes of the half to give the fans something to excite them.
That earlier strike had clearly encouraged Stevenson, the left-back latching onto a loose ball to fire in a low shot which caught the heel of Souttar and flew wide with Zlamal throwing himself to his left behind him three minutes after the restart.
Hibs could have taken the lead in the 57th minute when Stevenson tricked his way past two opponents and delivered an inviting cross for Vykintas Slivka. But the Lithuanian internationalist, who had replaced Flo Kamberi at the interval, sent his close-range header straight at Zlamal.
And they certainly should have three minutes later when Ikpeazu brought down Hanlon as he attempted a clearance inside his own penalty area, referee Craig Thomson immediately pointing to the spot. McNulty, who had scored three of his seven previous goals from the spot, stepped up – going for power rather than placement – and hammered a wild shot well wide of the target from 12 yards.
But Hibs did get their noses in front in the 69th minute, Slivka and David Gray combining to create the space for Horgan, the scorer of both goals in their win at Tynecastle a few weeks ago, to drill in a low cross which clipped Berra and went his own net.
Hearts, though, weren't out of it and hauled themselves level six minutes from time. MacLean, one of three second-half substitutes made by Levein, found space on the Hibs right to slide a low ball across for Ikpeazu to drill beyond Marciano.
The Jambos suddenly sensed a winner and Marciano had to be alert, rising to touch over an overhead attempt by Ikpeazu before substitute Ryan Edwards almost caught the Israeli goalkeeper as he made a clearance from McGregor’s short backpass.
Hibs: Marciano, Gray, McGregor, Hanlon, Stevenson, Mallan, Milligan, Omeonga, Horgan (Agyepong 89), McNulty, Kamberi (Slivka 46). Substitutes not used: Bogdan, Bartley, Gauld, Spector, Shaw.
Hearts (3-5-1-1): Zlamal; Souttar, Berra, Shaugnessy; M Smith, C Smith (Edwards 65), Djoum, Cochrane (Bozanic 71), Mulraney; Clare (MacLean 69); Ikpeazu. Substitutes not used: Doyle, Lee, Brandon, Dikamona.
Referee: Craig Thomson.