The hosts got off to a miraculous start, taking a double-digit lead already in the first quarter behind terrific play from Andrew Warren, who scored 13 points in the opening frame.
Having led by as many as 16 points in the second period, Port of Parnu looked to be fully in control of the game, despite seeing their lead dwindle down to 46-37 heading into the halftime break.
Nevezis had closed the gap down to four points in third quarter, but the hosts went on another run to restore the double-digit lead. However, a 16-0 run for the Lithuanians turned the game on its head, with Nevezis turning an 11-point deficit into a 74-69 lead with four minutes remaining.
Trailing by six points with half a minute left, Port of Parnu finished the game with a quick 7-0 run, crowned by a three-pointer by Warren, who finished the game with 30 points, to give the Estonians their first victory.
Paulius Petrilevicius added 21 points and eight rebounds, while Silver Leppik was the only other Port of Parnu player in double figures with 10 points, 12 assists and seven steals.
For Nevezis, Andrius Aleksandrovas and Mindaugas Kupsas shared a team-high 16 points, with the latter also adding nine boards. Taylor King collected 12 points and eight rebounds.