Lions opened the scoring but Mark Moore made it 1-1 before the half-time break.
In the second period, Haxby again took the lead and seemed to be heading for victory, but Liam Stilgoe struck a late equaliser to take the tie into extra time.
Gary Walgate put Leeman ahead for the first time before Moore notched his second ahead of a Neil Hartley goal to complete the triumph.
With goals from Charles Wrigley and James Shimmin, Enterprise saw off Albion Vaults 2-0.
In-form FC Barlby go into the hat after two goals from Louis Blackwell, plus strikes from Lee Carter and Tom Nicholson, underpinned a 4-2 defeat of Drax.
In division four of the Ian’s Cars of Barlby Sunday League, Selby Railway beat F3 6-0.
Railway opened the scoring with a volley from Kevin Weston. John Teale made it 2-0 with a direct free-kick before Jan Takacs rounded off a solo run with a classy finish.
In the second half, the Selby side increased their lead when James Hanson struck before prolific marksman Josh Knowles struck twice to take his season’s tally to 16 goals in four games to round off the emphatic victory.