The Bulldogs' top-eight hopes appeared in peril when Melbourne, in their first game since coach Simon Goodwin was sacked, chased down a 21-point deficit to lead by 10 at three-quarter time.
But Beveridge's charges found an extra gear to prevail 15.15 (105) to 15.9 (99) in front of 34,733 fans at the MCG on Sunday.
"We've had some monumental wins over the course of my time with the club and this is one of the best. It really is," he said.
"Because at three-quarter time, the game had gone south.
"... I just think it's outstanding. It's really important for us."
The Bulldogs (52 points) sit ninth, four points behind Gold Coast, Hawthorn and GWS, with home games against West Coast and Fremantle to come.
"I still feel like if we can win the last two, we might be able to sneak in," Beveridge said.
Melbourne's seven-goal third quarter put them in the box seat at three-quarter time.
The final quarter featured five lead changes, but the Bulldogs won out, with Sam Darcy, well held by Tom McDonald all day, taking an incredible late mark going back with the flight.
"What do they say? 'It might not be your day but it might be your moment' - well he had a couple of critical moments," Beveridge said.
The magnificent Ed Richards had 31 disposals and 14 clearances while kicking two crucial final-quarter goals among 15 score involvements.
Marcus Bontempelli (33 disposals, seven clearances) was crucial while Aaron Naughton (five goals) led the attack.
"Our midfield group took control of centre bounce at different stages, which was absolutely critical and we pushed over the line," Beveridge said.
"... Ed Richards, Marcus, and Libba (Tom Liberatore) just had enormous contributions when it really mattered at the end there.
"Obviously Naughts (Naughton) has just had a magnificent day, he's such a Goliath down there at the moment and Westy (Rhylee West) stood up late in the game.
"So they really put their flag in the ground. And as I said, we fight on. We're still alive."
Demons skipper Max Gawn (42 hitouts, 23 disposals) was superb while Clayton Oliver (27 disposals) was busy.
"It's been a tough week for everyone, but for the guys to play out the way they did, and to be so close, I think it says a lot about this group and the path they're on," caretaker coach Troy Chaplin said.
"I thought our leaders were fantastic ... Against a team who are in really good form and pushing for finals, (to lose) by a goal and right in the last minute, I couldn't be prouder of the group."
Chaplin "really enjoyed" his first game in charge, with matches against Hawthorn and Collingwood to come, but reiterated he won't go for the full-time role.
Sydney premiership coach John Longmire ruled himself out of contention, with Nathan Buckley and Adam Simpson apparent frontrunners.
Demons football boss Alan Richardson, who expects to remain in his role, told ABC radio Melbourne had not spoken to Buckley.
Pre-game, injured defender Jake Lever stressed his commitment to Melbourne and believed an incoming coach would be excited by their list.
Lever also told Triple M radio he wants to get to the bottom of how Goodwin's sacking leaked before most players were told.
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