Christian Nade headed off the bar in an open first half, and both he and Gary Glen were denied as Hearts pushed forward after the break.
The visitors also had chances and accomplished displays by goalkeeper Yves Ma-Kalambay and holding midfielder Sol Bamba helped them take a point.
Hibs boss Mixu Paatelainen employed an ambitious 4-3-3 formation with winger Alan O'Brien in midfield, but Bamba made up for their numerical disadvantage as he swept up countless times in front of his back four.
Jonatan Johansson had a quiet debut while Derek Riordan and Lewis Stevenson were recalled, and Steven Thicot replaced the injured Rob Jones in central defence.
Jason Thomson and Eggert Jonsson came into the Hearts defence in place of the suspended Marius Zaliukas and Lee Wallace.
A full house generated a great atmosphere as the teams ran out but it turned sour when some Hibs fans interrupted a minute's silence for former Hearts captain George Miller, who recently died aged 69.
The game opened at a frenetic pace and Ma-Kalambay soon beat Andrew Driver's powerful shot past his near post.
Christophe Berra produced a crucial block on Chris Hogg at the other end after Riordan's free-kick into the box.
The first real threat came in the 16th minute when Nade headed Robbie Neilson's long throw off the bar.
The visitors created their first serious chance midway through the half when Stevenson laid the ball off for Fletcher, but he blasted over from 16 yards.
Riordan shot across the face of goal as Thomson was off the park receiving treatment for a head knock.
The former Celtic player soon had another chance after Fletcher was fouled 30 yards out, but Janos Balogh got down comfortably to hold his curling free-kick.
Ma-Kalambay made the save of the half when he touched wide Laryea Kingston's powerful strike from the edge of the box.
Thomson, who was injured when he slid off the pitch after challenging David Van Zanten, was taken off in the 34th minute. Teenager Glen came on in midfield with Christos Karipidis dropping to centre-back and Jonsson moving across to left-back.
Glen almost lobbed Ma-Kalambay after Hogg misjudged a through ball, but the Hibs keeper got his fingertips to the ball.
Hearts put their rivals under sustained aerial pressure but they needed a perfectly-timed sliding tackle from Jonsson to prevent Riordan going clear from Stevenson's through ball on the counter-attack.
Ma-Kalambay got down well to save Michael Stewart's shot in the closing seconds after he had worked space just inside the box.
Nade struck the woodwork again five minutes into the second half. Jonsson surged down the left and cut the ball back for the former Sheffield United frontman, whose left-footed shot bounced off the outside of the post.
Deividas Cesnauskis replaced Kingston in the 54th minute and, after Hearts survived a spell of pressure, he was involved in a decent move that created a shooting chance for Glen, whose low effort on the turn was smothered by Ma-Kalambay.
Hibs came close when Fletcher flicked Riordan's free-kick inches wide of the post, but the woodwork came to their rescue again in the 64th minute.
Glen took the ball from Stewart 30 yards out and unleashed a dipping strike which flashed off the top of the bar.
Referee Calum Murray was helping the game flow but he eventually handed out his first yellow card for Stevenson's unsuccessful attempt to take down the breaking Ruben Palazuelos.
Hearts were having the better of the play but they were growing increasingly desperate in their finishing, especially when Nade leaned back and fired well over from 15 yards following Neilson's low cross.
Fletcher nearly hurt them on the break after cutting inside Neilson following a clever pass from sub Shiels, but Balogh blocked his close-range effort.
Cesnauskis cut inside and fired just over from 20 yards before Riordan was booked for a wild tackle on Driver.
Berra and Ian Murray were also booked after clashing off the park as Hearts threw men forward, but Ma-Kalambay again looked secure as he parried the Hearts captain's powerful strike from just inside the box.
Bamba hooked over after Fletcher's header was blocked by Neilson at a set-piece as Hibs threatened to steal the points late on.