London Fire Brigade (LFB) sent crews from Shadwell, Shoreditch, Bethnal Green and Poplar.
It said the cause of the fire was under investigation.
Rob Lane, chief property officer for Clarion, which is responsible for the block, said: "Our staff have been at Bradbeer House since the early hours of this morning supporting residents and organising emergency accommodation.
"Our priority right now is our residents' wellbeing and we have teams supporting them.
"We thank the LFB for their efforts to extinguish the fire and the council for their support and are working closely with them and other key partners to understand what has happened and get residents back to their homes as quickly as possible."
A London Ambulance Service spokesperson said: "We sent a number of resources to the scene including ambulance crews, advanced paramedics and members of our Hazardous Area Response Team.
"We treated 17 patients in total. We took 11 patients to hospital and six patients were cared for and discharged at the scene."
Tower Hamlets Council said a rest centre had been established to provide immediate support for residents.
A spokesperson said: "Vulnerable residents received dedicated care, and alternative accommodation, including hotel placements, was arranged for those who were displaced."
They added that the housing association for the building, Clarion Housing, would lead recovery efforts but the council would "provide support where needed and assist in ensuring the safe return of residents to their homes".
