Team GB picked up gold in the same event at the last World Championships in Kazan in 2015 when Scott was part of the team.
They then earned a silver in the Olympics in Rio last year – with Milne and Scott involved – and now they have ended up on the podium once again.
It means that Milne and Scott are the third Scots to pick up a medal at the overall aquatics meet, diver Grace Reid from the Edinburgh Diving Club having won a silver medal with Tom Daley in the 3m mixed synchro event last Sunday.
In the heats in the morning yesterday, Milne, the 23-year-old City of Perth swimmer, swam the first leg in 1:46.78.
Nick Grainger and Calum Jarvis then swam the second and third legs before handing on to Scott, the 20-year-old from the University of Stirling, who brought the team home with a final leg of 1:46.50.
That gave them a time of 7:05.79 to finish second in the heat behind Australia and safely make it through to the evening session in Hungary.
In the final, the Great Britain team was re-jigged slightly with James Guy coming in for Jarvis and Scott moving back to the third leg so that Guy could swim the fourth leg.
Milne got the team off to another solid start with a time of 1:47.25. Grainger then went well, but at that stage the USA and the Russian Federation were well ahead.
The Brits knew that they had their best two swimmers to come and Scott did an excellent job in closing the gap between himself and the leading teams.
Scott’s time was 1:44.06 and then Guy just powered into the water and took the race away from the opposition.
His time of 1:43.08 was superb and Milne, Grainger and Scott were cheering him home the whole way.
Their overall time was 7:01.70, with the Russian Federation second in 7:02.68 and USA third in 7:03.18.
Milne, who finished 15th in the 400m freestyle earlier in the week, said: “It is such a good feeling right now.
“I knew I just had to get out and give the team a solid start. It was nervy watching the rest of the race unfold, but Nick, Duncan and James did really well and it is always special being part of a team that has success. The guys on the podium want to give credit to Calum Jarvis, who swam in the morning too.
“For me personally this is such a good feeling and sets me up well for the future.”
For Scott, the medal made up for a couple of near misses earlier in the meet.
On Tuesday, he finished fourth in the 200m freestyle, losing out on a medal by 0.04 seconds. And on Thursday he was well in the mix for a medal in the 100m freestyle right up until the last few metres, just missing out in fifth with a time of 48.11.
“I was feeling a bit down on Thursday night after the 100m freestyle, but I knew I had to switch on quickly to the next job and when you are part of a team with guys like this it is easy because they pick you up,” he stated.
“It also helps when you know you have good form in the event and after the heats we were feeling in a good place.
“Stephen and Nick took the team out well and myself and James were just glad that we could reel the Americans and the Russians in.
“This makes up for my personal close losses 100 per cent.”
Meanwhile, Ross Murdoch was just edged out of a medal in the 200m breaststroke final.
The 23-year-old from the University of Stirling swam 2:07.72 in his semi-final on Thursday night, but this time posted 2:08.12 to finish fourth.
Anton Chupkov of the Russian Federation won gold in 2:06.96, with Yasuhiro Koseki second and his Japanese teammate Ippei Watanabe third.
Murdoch, who was 0.65 seconds behind Watanabe, said: “It is disappointing because I was feeling good and felt I had swum the race well tactically, but it wasn’t to be.”