Brentford were promoted to League One as Champions after beating Darlington 3-1 on Saturday.
The Bees' task was made easier when the Quakers were reduced to ten men within two minutes of kick-off with Alan White's dismissal for violent conduct.
Alan Bennett then put the visitors in front in the 35th minute and Billy Clarke headed home two minutes before the break to double the advantage.
Clarke then scored in the 54th minute to seal the victory for Andy Scott's side, although the Quakers got a consolation via Pawel Abbot on 59 minutes.
But there is still all to play for behind the Bees as Wycombe, Exeter and Bury remain in the hunt for the last two automatic promotion places after registering draws.
Wycombe remain second after coming from behind to draw 1-1 at Port Vale.
Lee Collins put Vale in front in the 61st minute but Lee Sawyer got the equaliser for Peter Taylor's side with ten minutes to go.
Exeter are in third following a 2-2 home draw with Morecambe.
Stuart Drummond headed the Shrimps in front (9) before Dean Moxey (49) equalised after the break. Ryan Harley gave Exeter the advantage four minutes later, but Drummond had the last word with a 79th minute equaliser.
Bury remain in fourth - three points behind Wycombe and one behind Exeter - following a goalless draw at Gillingham.
The Gills remain in fifth - and are guaranteed a play-off place with the point - but this result ended their hopes of automatic promotion.
Rochdale stay in sixth - and are also guaranteed a place in the play-offs - after coming from behind to draw at Luton.
Tom Craddock put the relegated Hatters ahead in the 33rd minute from the penalty spot before Adam Rundle got the crucial equaliser in the 56th minute.
Dagenham maintained their grip on the last play-off place with a 3-0 win at Notts County.
Sam Saunders netted in the 34th minute and Matt Ritchie scored a brace (44 & 88) to keep the Daggers in seventh.
Shrewsbury are now two points behind in eighth after being held to a goalless home stalemate by Lincoln.
Mid-table Accrington ended Chesterfield's play-off hopes with a 1-0 victory courtesy of Robert Grant's 50th minute winner.
The Spireites' day then got worse when top scorer Jack Lester was sent-off for foul and abusive language.
A Peter Thorne brace helped Bradford arrest their slump and beat Rotherham 3-0 in the Yorkshire derby but their slim play-off hopes still ended because of Dagenham's win.
Thorne netted on 12 and 32 minutes, while Steve Jones made sure of the win on 73 minutes but the win was too little-too late for Stuart McCall's side.
Down at the bottom, Chester are all but relegated after drawing 2-2 at Aldershot.
Mark Wright's side are three points behind Grimsby with one game to play but have a vastly inferior goal difference - 19 goals - to the Mariners.
Ryan Lowe and Lloyd Ellans scored for Chester, while Kirk Hudson netted a brace for the Shots.
Grimsby lost 2-1 at Bournemouth after the Cherries came from behind to claim the win that guarantees their league status.
The Mariners went ahead in the 41st minute via Nathan Jarman but then had Danny Boshell sent off on the stroke of half-time for a second bookable offence.
The Cherries took advantage of their numerical advantage when Liam Feeney equalised two minutes after the re-start and Steve Fletcher netted the winner with ten minutes to go before the Mariners' Joe Widdowson was sent off in stoppage time for a second yellow card.
In League Two's last game Macclesfield beat Barnet 2-1 at the Moss Rose.
Nat Brown put the home side in front in the second minute before the Bees equalised via Albert Adomah (54).
But substitute Kristian Dennis scored with eight minutes to go to give the Silkmen the win.