It meant that Wanderers retained their undefeated record and, with two matches still to play, are guaranteed a deserved step-up to Championship 1 in the second half of the season.
Exeter were 7-3 behind at one stage but briefly led 10-9 in the third before starting the final, tense period at 10-10. Sam Alberici struck early blows for the Devon side to go 12-10 ahead, soon equalised by two goals from Joe Jenkins-Delf. There were just 65 seconds remaining when a Tom Hutchens goal put Exeter ahead again - only for Laxton to have the final word.
Sutton & Cheam are also through to Championship 1, staying unbeaten with a 9-5 victory over Watford. Watford led 7-6 at one point, but a third period hat-trick by Alasdair Wakeling - and a match tally of five goals from Ryan Symington - led to a 9-5 victory.
Portobello made short work of Scottish rivals Caledonia, scoring eight times in the third period on the way to a 17-5 win. Chief goal scorers were Angelos Mongolias (4), Neilson Rutherford (3) and Ben Scott (3). City of Bristol won for the first time - 15-10 against Penguin. The teams were locked at 10-10 with five minutes left, but Bristol produced a strong finish with goals by Elliott Derrick (2), Matt Thresher, Nathan Brown and Tom Whiting.
At Walsall, Sheffield Dolphins beat Invicta Marlins 5-4 to finish top of men's Division 5. Both qualify for Championship 4 along with Watford Saxons, the Army and Leeds, who beat the Saxons 11-5.