Heavy rain and a strong wind meant conditions were difficult for both teams, and this was borne out as both sides took a while to settle with Hearts having the first chance after 18 minutes when Phil Stamp saw his volley just scream past the far post from 16 yards.
It took 23 minutes for Thistle to mount a serious attack when Juan Ramon Escalas and Armand One combined well, but One's dangerous-looking cross was cut out and cleared for a corner.
Thistle had the next chance aftern 26 minutes when Derek Fleming's quick thinking saw Craig Gordon have to make a save from his 16-yard free-kick.
Neither side were able to produce any other clear chances until the 36 minute when Escalas saw his 18-yard shot hit the post before being cleared to safety.
The second half was well under way before either side had any chances, and Hearts' Robbie Neilson should have done better than see his prodded shot go wide from 14 yards. Hearts continued to force the game, and Stamp, Ramon Pereira and Kevin McKenna all saw their chances foiled either by sharp defending or poor finishing.
Hearts pushed for the winner, but despite their best efforts Thistle stood firm and deserved a replay a week on Wendesday.