He narrowly missed out to Italy's Nicolo Martinenghi who took gold.
Adam achieved the same time as American Nic Fink of 59.05 seconds, with both awarded silver.
Both Adam and the American were just two hundredths of a second behind the winner and Peaty claiming a third consecutive men's 100m breaststroke title.
The Briton said: "It has been a very long way back. I look at the score there and you would never think in your wildest dreams that it would be 59 seconds to win it.
"I gave it my absolute all. I executed it as well as I could. It's not about the end goal, it's about the process. It doesn't matter what time it says on the scoreboard. I think in my heart I have already won."
The emotional swimmer, 31, added: "I'm so happy that I can race against the best in the world and still come joint-second. These are happy tears. I gave my absolute best every single day and I cannot be upset about that."
He had a tearful reunion after the race with his three-year-old son George and partner Holly Ramsay.
Earlier Sir Andy Murray and his doubles partner Dan Evans showed true grit to battle through to the second round after a thrilling match.
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