Heracles' only previous league victory this season came in a 6-1 mauling of NAC Breda at the start of October, but they doubled their points tally at De Adelaarshorst on Sunday.
Marnix Kolder broke the deadlock for the hosts in just the eighth minute, only for Deniz Turuc to receive a straight red card for a lunging tackle on Fahd Aktaou in the 21st minute.
Linssen - who had not scored in 10 previous league matches this term - then levelled two minutes after the restart.
His second arrived in the 67th minute and he wrapped up the win in the third minute of stoppage time, as Heracles jumped above Dordrecht in the table.
Tjaronn Chery's second-minute strike was enough for Groningen to beat out-of-sorts NAC in a 1-0 win.
NAC have now failed to win their past six Eredivisie matches and they still occupy the final relegation play-off spot.
AZ and Excelsior played out a thrilling 3-3 draw at the AFAS Stadion. The hosts overturned a 2-1 deficit only to throw away the lead when Carlo de Reuver grabbed an 84th-minute leveller.
In other matches, Utrecht defeated Vitesse 3-1, while Samuel Armenteros' last-gasp goal earned Willem II a 1-1 draw at home to Cambuur.
