Ronald Koeman made a big call by bringing Brobbey into his starting lineup in place of Crysencio Summerville, who scored against Japan, but that decision was vindicated by two goals inside the opening 17 minutes.
First, Brobbey took down a long clearance and laid the ball off to Tijjani Reijnders, who sent Cody Gakpo away down the left, and his low centre was fired in from six yards by the Sunderland striker.
Sweden came close to an immediate response when Viktor Gyokeres forced Bart Verbruggen into a smart stop at his near post, only for Brobbey to add his second and the Netherlands’ 100th World Cup goal by diverting another low cross - this time from Denzel Dumfries on the right - past the helpless Kristoffer Nordfeldt.
The first-half hydration break gave Sweden a much-needed chance to regroup, and they were better coming forward after the restart, stretching the Dutch defence, but Yasin Ayari was guilty of a lack of cutting edge in front of goal more than once.
In the final 10 minutes of the half, Gyokeres swept a low right-footed effort goalwards from the left side, before Gakpo produced similar effort at the other end, but Bart Verburggen and Nordfeldt were equal to them.
The Oranje were almost out of sight when Donyell Malen got free only to flash his strike across goal, but the end-to-end nature continued when Ayari fired narrowly over from 18 yards.
Sweden thought they were right back in when Gustaf Lagerbielke nodded in Benjamin Nygren’s free-kick, but it was ruled out for offside, before Verbruggen palmed away a direct free-kick from Gyokeres and a clean hit from Ayari on the stroke of half-time.
All the Swedish endeavour to get back into the game was undone within two minutes of the restart, and again the Dutch used a low ball across goal to punish them.
Dumfries squared the ball down the corridor of uncertainty, and though Brobbey couldn’t connect, Gakpo was there to bundle it home.
The Liverpool man needed just seven more minutes to join Brobbey on a brace, as substitute Summerville burst through the middle before feeding Gakpo on the left, and his fierce strike at the near post was too quick for Nordfeldt.
Those goals still didn’t quash the belief of Graham Potter’s side, as Ayari went close from range before they finally got on the scoresheet, when Alexander Isak sent Anthony Elanga through, and the replacement finished coolly past Verbruggen.
A win by the same scoreline over Tunisia in their opener keeps Sweden in with a chance of reaching the Round of 32, a chance they could have strengthened with further consolations, but Verbruggen kept out their best effort in the final 10 minutes - a curling strike from Isak - before Summerville ensured it was back-to-back 5-1 scorelines for the Swedes by firing into the bottom corner from 20 yards.
It’s 13 games without a clean sheet for Sweden, who have now won just one of their last eight against fellow Europeans at the World Cup.
Excluding penalty shootout defeats, the Netherlands are now unbeaten in 14 World Cup matches since defeat in the 2010 final, surpassing the previous record set by Brazil between 1958 and 1966.
Flashscore Man of the Match: Cody Gakpo (Netherlands)