The quartet of Danny Talbot, James Ellington, Andy Robertson and Richard Kilty crossed the line in 38.21 seconds to set a new championship record.
The run is the fastest the team have done outside of a major championships in recent years and helped Great Britain and Northern Ireland to fourth place overall with 166.5 points from the first day.
And Richard Kilty, who earlier finished second in the 100m with 10.35, believes the squad did exactly what they were capable of against the best the continent has to offer.
“My aim was just to go past those on the outside and I managed to do that – we were too good for them,” said the World and European Indoor 60m champion.
“It’s a massive improvement from the Bahamas and we’ve all come a long way since then and it’s a good little squad we’ve got here.”
There was more record running as Eilidh Child set another European leading time in the 400m hurdles where she clocked 54.46 for victory.
Child set times of 54.84, 55.14, and 55.97 in this month's Diamond League events but bettered all those with a determined run in Russia.
“I just went out there to exercise my stride pattern, I wanted to win the race obviously to get the points for the team,” she said.
“I knew it felt like a nice track yesterday when we came in to warm up, but I knew it was pretty windy so I wasn’t too bothered about the time - so to win it and to knock a chunk off my season’s best I’m delighted and hopefully this can set me up for the next few weeks now.”
The day started with a bang for the Brits as Asha Phillip stormed to 100m victory in 11.27 into a 1.4m/s headwind.
And that was followed by further victory in the men's 400m where Jarryd Dunn clocked a new personal best of 45.09 to take maximum points for the team.
“I’ve raced three 400s this year and that was my third personal best – I’ve PB’d at every one of them, so that’s great,” said the Birchfield Harrier.
“I would have liked the time to have been a tenth quicker than that like 44.9 but I’m going in the right direction given that six weeks ago my PB was 46.00 from 2012!”
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