The Russian and US quartets were neck-and-neck after three laps, but Russia's Antonina Krivoshapka powered ahead of Francena McCorory to send the home fans into a frenzy of delight.
Ohuruogu took the baton for the final leg for Britain after Margaret Adeoye, Eilidh Child and Shana Cox had combined to leave her neck-and-neck with the French team, but around 25m behind the leading pair.
Gold from so far back was too much to ask for the newly-crowned world 400m champion, but Ohuruogu never looked in any danger of missing out on bronze to collect her second medal of the championships and the fifth World Championships medal of her career.
The medal for Britain was followed just minutes later by another as Tiffany Porter earned a brilliant bronze for Britain in the women's 100m hurdles as USA's Brianna Rollins took gold.
Rollins shocked Australia's Olympic champion Sally Pearson to win in 12.44 seconds, with Pearson second on 12.50 and Porter clocking 12.55 to come third.
Porter's bronze gave Britain its first ever World Championship medal in the 100m hurdles.
In the final race of Saturday's action, British sprint star Adam Gemili came agonisingly close to earning a third bronze as he came off the bend in the men's 200m final in third place and looking strong.
Yet the 19-year-old was overtaken in the final stetch to finish fifth in 20.08 seconds.
It was a fine effort for such a young sprinter, but he will no doubt be disappointed to have run a tenth of a second slower than he managed in the semi-finals - particularly given that his time of 19.98 on Friday would have been good enough for bronze.