Yarisley Silva of Cuba won silver with the same height and twice former champion Yelena Isinbayeva of Russia took bronze after only clearing 4.70m, 36cm below her own world record.
Bleasdale had been tipped for a medal but appeared to struggle under the pressure and came sixth, managing 4.45m, well below her personal best of 4.71m.
Suhr, silver medallist behind the Russian world record holder in Beijing four years ago, cleared 4.75 for victory on a cool overcast evening.
Cuba's Yarisley Silva took the silver with the same height and Isinbayeva had to settle for bronze with a best of 4.70.
Isinbayeva lacked her usual authority in the event, failing the first attempt at her opening height of 4.55.
In contrast, Suhr coped better with the conditions and piled on the pressure, needing just one vault at each of her heights until 4.75 which she cleared second time.
After the 30-year-old Isinbayeva hit the mat for the last time having knocked the bar off with her thigh, she got up with a big smile on her face, waved to the crowd and blew a kiss to the camera.