Membrey marked about 45 metres out from goal but his attempt at a match-winning goal flew agonisingly to the right, with the Bulldogs prevailing 14.12 (96) to 14.9 (93) in front of 81,036 fans at the MCG on Friday night.
The Dogs advance to an elimination final next week, while the Magpies' finals campaign is over.
Beveridge's men led by as much as 21 points in the third quarter but had to ride their luck when a Nick Daicos-inspired Collingwood surged back into the contest.
There were four lead changes in time-on of the final quarter before Membrey's critical miss settled a messy, yet enthralling contest.
"There was a fair bit of green on the stats sheet but we know Collingwood just keep coming and it was a fairly ballistic game in the end," Bulldogs coach Beveridge said.
"The heart's settled down but when there's that many lead changes late in the game and you're torn between where your numbers should be, it's difficult to get your calculations right.
"Some of the late centre clearances were pretty crucial and (Aaron) Naughton and Arty (Jones) were clutch at the end there being able to put us back in front twice.
"It never felt comfortable and ultimately both teams made mistakes ... a bit relieved, obviously."
Membrey put Collingwood in front at the 26-minute mark of the last quarter but Jones hit back quickly with what ultimately proved to be the decisive goal.
Beveridge noted post-match that Membrey's shot clock had run out before he started his run-up for the final shot, and should have been called to play on.
The siren then sounded half-way through Membrey's run-up and Beveridge admitted there was "a little bit of luck" involved in his side's triumph.
Magpies teammates and opponents rushed to console Membrey after his miss.
"We're not in that position without that big mark and big goal (from Membrey) one minute before," Collingwood coach Craig McRae said.
"It was a really see-sawing game.
"It's hard not to, and I can't imagine what that would be like, but he (Membrey) doesn't need to carry the weight of that (miss). That's not all his."
Raging Brownlow Medal favourite Daicos (30 disposals, two goals) did everything he could to drag Collingwood over the line in a battle of the two marquee men with opposing skipper Marcus Bontempelli.
Bontempelli (32 disposals, one goal), Matthew Kennedy (27 touches) and Ed Richards (25) gave the Bulldogs midfield ascendancy, while Naughton and Jones fired in attack with four goals each.
Kennedy was hurt in a final-quarter collision and was ruled out of the latter stages of the match - as well as next week's elimination final - with concussion.
Bailey Dale (24 disposals) was influential at half-back.
Dan McStay kicked four goals for Collingwood, with youngster Sam Swadling (32 disposals) and Josh Daicos (26) also busy, and Angus Anderson kicked two important goal in the final quarter
The Magpies' defeat brought down the curtain on premiership hero Steele Sidebottom's career, with doubt over injured games record holder Scott Pendlebury's playing future.
The Bulldogs will find out who they play in next week's elimination final after Saturday night's other wildcard final, between Melbourne and Carlton, also at the MCG.
