However, Lynn took the lead four minutes into the second half when defender
Mark Burrows headed a Paul Raynor corner firmly into the net.
As Burnham lost their discipline, Staff fired a free-kick just wide.
The hosts' best chance of the game came on 54 minutes as Michael Bartley pushed the ball past Greygoose, but it rolled inches wide of the opposite corner.
But Lynn regained control of the match and it was no surprise when they scored after 72 minutes.
Great play from Bacon, who switched to Staff, whose cross resulted in a bout of head tennis, Stephenson gaining the final touch for his 19th goal of the season.
A minute later Burnham captain Paul Brett earned a red card with a lunge on Bacon.
The resulting free-kick from Raynor produced a brilliant save by Honey, who had also somehow kept out a Defty thunderbolt.
With a quarter of an hour remaining, Grey-goose got a kick while on the floor. He soon fumbled a cross, and was clearly in distress.
He was replaced by Dean Mayes, who was soon in action, clutching a shot from home sub Alex Toley.
The Linnets went close before sealing the game in injury time, Paul Watts finishing another Staff centre at close range not long after coming off the bench.
Lynn: Greygoose (Mayes 83), Harvey, Cowling, Burrows, Fuff, Defty, Weston, Raynor, Bacon (Watts 88), Stephenson (Ortega 90), Staff. Unused sub (Blois).
Burnham: Honey, M Vrhovski, Brett, Bracken, Mitchell, Lockhart, Bryson, Ferdinand, Lennon (Brown 77), Bartley, Davies (Toley 77). Unused subs J Brown, S Brown, Nutter.
Referee: M Ives Att: 137
Bookings: Lynn - Fuff (60, foul). Burnham: Lockhart (48 kicking ball away), Ferdinand (53 foul).
Sent off: Brett (73 foul). Lynn News Men of the Match - Jack Defty/Chris Bacon.