The 24-year-old finished in joint third position with two other athletes by clearing 2.29m but failing to make it over 2.33m.
Speaking to BBC Sport, Grabarz said: "I got a medal and I'm over the moon. I'm so happy.
"It wouldn't have been possible without my mother and my coach. If you told me when I was young that I could have a bronze medal at an Olympics in London I'd have bitten your hand off."
Russia's Ivan Ukhov took gold by clearing 2.38m, with Erik Kynard of the United States claiming silver.
Grabarz's triumph saw Great Britain move to 48 total medals, one more than the number collected in Beijing.