The Stirling-based swimmer now has six medals overall – with just cycling trio Chris Hoy, Jason Kenny, and Bradley Wiggins having more.
Speaking after the history-making swim Scott said: “It’s all my teammates, that’s what that is. I’m fortunate to be part of some excellent relay teams, this being one of them.
"We’re all pretty disappointed with this but in 2015, when we started coming together, back-to-back Olympic silvers in this event... we’d have taken that all day.
“We're coming in here to really challenge the Americans just like we did last time at Worlds. With our splits, we all swam well, slightly disappointed with mine but that’s how it is. It’s been a tough week, but we’ve got to be happy with that.”
The Team GB quartet of Scott, Luke Greenbank, James Guy, and Adam Peaty finished 0.73secs behind winners America, who set a new world record time of 3:26.78 as Italy took bronze.
Silver means Scott has won four medals at a single Games, setting a new British record.
The result also means GB sign off from Tokyo 2020 having enjoyed their most successful stint in the pool at an Olympics with four golds, three silvers, and one bronze.
At London 2012 the swimmers managed just three medals – one silver and two bronze – and won just one gold at Rio in 2016.
American swimmer Caeleb Dressel also joined an elite club with a fifth gold in the pool. Only East Germany’s Kristin Otto and Dressel’s compatriots Michael Phelps, Mark Spitz, and Matt Biondi have managed the same feat.
Before winning a team gold in the relay with the USA, Dressel won gold in the men's 50m freestyle.
The 24-year-old set a new Olympic record time of 21.07secs and added to his 100m freestyle, 100m butterfly and 4x100m freestyle relay titles in Tokyo.
Britain's Ben Proud finished fifth in the race, missing out on a medal by less than two tenths of a second in a tigher-than-tight finish as France's Florent Manaudou took the silver and Bruno Fratus of Brazil the bronze.