The wasteful Lions led by 17 late in the second term before the Demons kicked five consecutive goals in the third quarter to set up a show-stopping finish.
Melbourne took the lead in the fourth, then withstood a late Lions surge to claim a memorable 16.8 (104) to 15.12 (102) win at the MCG.
"If you want to step in the ring with the big boys, you've got to put score on the board," King said.
"I think even today, if we went defensive for a while, Brisbane's talent would have got us in the end.
"So we had to try and win the last quarter. We couldn't hold on."
Melbourne have beaten Carlton, St Kilda, Gold Coast and Brisbane.
When asked if Sunday's win was their best, King answered: "I think so".
"Six games in, when you looked at the draw and saw Brisbane on it, you'd hope you compete, and you'd love a win," he said.
The result condemned 300-game Lions champion Lachie Neale and 200-gamer Josh Dunkley to defeat in their milestone games.
Melbourne's Harrison Petty was helped off at the end of the third term when he appeared unsteady on his feet and complained of blurred vision in a worrying moment.
King said the club was still investigating but the case could be one of delayed onset concussion.
"We're obviously going to take it pretty seriously and we'll do the due diligence to make sure he's OK," King said.
Kysaiah Pickett could come under investigation after Brisbane's Noah Answerth was concussed when a fourth-quarter bump from the star Demon sent the Lion into Brody Mihocek's shoulder.
Brisbane also lost Jarrod Berry to a second-quarter calf injury and Bruce Reville (ankle) limped off in the fourth.
"We didn't have too many fit guys on the bench, but we found a way to nearly pinch it," coach Chris Fagan said.
"So I thought there was good character shown by the group."
Second-year Demon Harvey Langford (27 disposals, three goals) was wonderful along with Kade Chandler (17 disposals, three goals), who sank two huge final-quarter majors.
Max Gawn (seven clearances, 23 hitouts, 21 disposals) worked over Sam Draper (23 hitouts).
Lions co-captain Harris Andrews (22 disposals, 12 marks) was masterful, though opponent Jacob van Rooyen (three goals) jagged two crucial final-quarter goals.
Neale (31 disposals, 10 clearances), Darcy Wilmot (27) and Reville were busy.
In his 50th game, Logan Morris dobbed three goals along with Charlie Cameron.
Lions ruckman Darcy Fort (illness) was a late withdrawal, with Zane Zakostelsky recalled.
Scores were level at quarter-time but Brisbane seized control in the second term.
The Demons were celebrating 'Jim's Game', commemorating late Irish club legend Jim Stynes, and had a violinist and Irish dance troupe performing a jig after every Melbourne goal.
That violin kicked into overdrive as Melbourne kicked five goals on the spin in the third term and led by six points at three-quarter time.
Hugh McCluggage edged Brisbane in front briefly but van Rooyen took a huge mark over Andrews to restore Melbourne's lead before Chandler sent the MCG into ecstasy.
The Lions mounted a late comeback and Rayner (2.3) missed a late chance to put Brisbane back in front before Melbourne held on.
"In the end, you can put it down to accuracy in a close finish like that," Fagan said.
Brisbane next host Adelaide at the Gabba on Sunday while Melbourne play Richmond at the MCG on Friday night.
