Osrin, making her debut for Great Britain, clocked a personal best time of 2:10.30 to make the podium in Russia.
Having only been fifth at the halfway mark, Osrin produced a stunning final 100m and it looked like she was going to win gold, but she was just edged out on the touch by Italy’s Erika Francesca Gaetani.
There was just 0.02 of a seconds between Gaetani and Osrin, with the Italian winning in 2:10.28.
“I’m really proud of her performance and hard work,” said her Leander coach Robin Armayan afterwards.
Osrin is due to swim in the 200m freestyle on Friday and the 100m backstroke on Saturday.
Meanwhile, fellow Plymouth Leander swimmer Sophie Freeman reached the final of the 200m butterfly at the championships.
The 15-year-old was consistent all the way through the three rounds, swimming 2:14.13, 2:14.27 and 2:14.29.
Freeman will also swim in the 50m and 100m butterfly events in Kazan.
Mount Kelly College’s Harry Noble, who came through Devonport Royal’s junior ranks, reached the semi-finals of the 100m backstroke in Russia.
He swam a PB time of 56.45 seconds in the heats and 56.69 in a strong semi-final.
He will go in the 200m backstroke on Friday, while Mount Kelly team-mate Oliver Taverner will open his campaign in the 50m breaststroke prelims on the same day.