Making her marathon debut in Barcelona, 28-year-old Tesfay finished well ahead of Kenya's Jepkosgei Kiplimo, who finished second nearly eight minutes behind. Zeineba Yimer was third.
"Today was fantastic, not what I was expecting, but it is good," Tesfay said with the help of an interpreter.
"My plan was to attack the world record, but today there was a lot of wind, and I was not able to push in the last part of the race.
"Today I did not succeed, but in the next marathon I would like to try for the world record."
Her time was 57 seconds short of Ruth Chepng'etich's record of 2:09:56, but it was the fastest debut ever.
The Kenyan was banned for three years in October after she admitted to anti-doping rule violations, but her world record mark will remain on the books as it was set before her positive test.
Uganda's Abel Chelangat won the men's race ahead of Patrick Mosin and Jonathan Korir.