He nutmegged Scott Robertson before firing home in the 76th minute to break the deadlock in the Edinburgh derby.
Robertson then brought down Prince Buaben to earn a second yellow card before the Hearts player got up to slot home the penalty.
Jambos striker Osman Sow was then sent off for elbowing Michael Nelson before Farid El Alagui pulled one back. Hibs had missed the chance to go in front when Liam Craig missed a penalty in the 31st minute.
Hearts head coach Robbie Neilson felt Nicholson's brilliance merited the points.
"Sam has quality. His shot was unstoppable. It gave us the impetus to win the game," he said.
Hibs head coach Alan Stubbs said of ref Willie Collum: "I'm not going to hide behind decisions but I think he missed one or two things."