The visitors made the running in the opening quarter at Tynecastle and Nadir Ciftci thought he had given them the lead but David Goodwillie had been penalised for a push.
Jamie Hamill's drilled shot forced Radoslaw Cierzniak into his first serious action before Danny Wilson also went close with a thumping header.
The home bench were incensed after Ryan McGowan saw his close-range effort ruled out, apparently for a foul on Cierzniak.
Jamie MacDonald had to be alert to deny the lively Ciftci as United countered and the Turk cracked one off the base of the Hearts upright moments later with a 16-yard effort.
Hearts kept pressing to their credit but even the introduction of talisman Ryan Stevenson - back after six games out - could not provide the Gorgie faithful with the goal they craved.