The hosts took the lead as early as the eighth minute when Marnix Kolder marauded into the box before firing home.
However, Foeke Booy's charges soon found themselves at a disadvantage as Deniz Turuc was sent off midway through the first half.
Go Ahead did well to take their lead into the break, but just two minutes after the restart, Linssen levelled proceedings.
The 24-year-old scored his second in the 71st minute, putting Heracles ahead when Mickey van der Hart's initial save was then poked home on the rebound.
And with injury time looming, Linssen then scored to confirm his hat-trick and guarantee the victory for his side - only their second of the season and one which lifts them off the bottom of the table.