Zhitnik received a kneeing major and a match penalty. But Hlavac managed to slide a pass to Jeff O'Neill as he tumbled to the ice and O'Neill put a one-timer past rookie goaltender Mika Noronen's stick side with 2:37 left in the first period.
Hlavac was playing his first game for the Hurricanes after being acquired Friday with Harold Druken in a trade with the Vancouver Canucks. He was helped off the ice and did not return.
The Sabres are winless in their last seven games, losing five during that stretch. After scoring 11 goals in two wins to open the season, they have just 15 in the last 11 contests.
Buffalo had a chance to tie it midway through the third period, but video replays showed the puck was pinned against the side of the net before it was kicked in by Vaclav Varada.