Sturrock's side came closest to scoring through new striker Bradley Wright-Phillips, who had a first-half shot on the spin spectacularly turned away by keeper Jamie MacDonald.
Late on substitute Rory Fallon headed over from Craig Noone's left-wing cross.
"In the second half you have got to give credit to Hearts, they came into the game and pressed higher up the pitch," said Sturrock.
"You have got to remember we were playing a team that was third to Rangers and Celtic last season - and they were a worthy third place."
Central defender Chris Barker, meanwhile, is to have a scan on a calf injury sustained in the warm-up before Wednesday night's draw.
Hungarian international Krisztian Timar was drafted in to partner Dutch defender Marcel Seip in the middle of the Pilgrims' back four as a result of the injury.
Sturrock said: "Chris pulled his calf in the warm-up and will have a scan.
"It's obviously disappointing but I thought Krisztian came in and did okay."