The visitors, who finished bottom of the Premiership last season, produced a stunning comeback in the final 25 minutes at Stark's Park.
Rovers took the lead in the 13th minute when referee Nick Walsh spotted a handball and Lewis Vaughan beat goalkeeper Ryan Fulton from the spot.
It was 2-0 in the 25th minute when fellow forward Dario Zanatta finished off a quick breakaway move by the home side.
Rovers were 3-0 ahead seven minutes into the second half when recent signing Aidan Connolly beat Fulton with a low shot, then Vaughan powered clear before coolly chipping a fourth in the 65th minute.
Accies' comeback started three minutes later when Kieran MacDonald headed in a cross from substitute Andrew Ryan, before Ryan himself made it 4-2 10 minutes later with a powerful angled drive.
MacDonald set up an exciting finish with seven minutes remaining when he turned in a low cross to make it 4-3, before Shaun Want capped the comeback with the equaliser two minutes into added time.