Fire crews from Barking, Dagenham, East Ham and surrounding fire stations were at the scene.
Roads were closed off to the public as emergency services worked to put out the fire.
The cause of the fire is not known and an investigation has been launched.
Social media users rushed to show their support for those affected by the fire.
Some suggested residents had previously complained that the use of wood on the flats was unsafe.
One person wrote: "My heart is absolutely shattered on seeing the fire in Barking Riverside - despite the residents complaining on the use of wood on the flats being dangerous."
Another person claimed that the company which built the block of flats had assured residents that the wood cladding was fire retardant.
Peter Mason, chair of the Barking Reach residents' association, told the Guardian that he contacted the builder Bellway Homes last month to ask for the fire risk to be investigated after BBC Watchdog reported claims of fire safety problems at two other developments.
He was told not to worry, according to an email seen by the newspaper, saying the construction method was different to the ones in the report.
A spokesperson for Bellway told the newspaper: "We are taking the matter very seriously and working with all parties to help in establishing the cause of the fire."
The blaze comes two years after the inferno at Grenfell Tower in North Kensington, which killed 72 people and left many more homeless.
Grenfell United, an organisation which supports those affected by the tragedy, tweeted last night: "Thoughts and prayers with everyone affected by Barking fire. We hope everyone is safe. So difficult to see this!
"Residents affected will need to be looked after & Council must offer safe accommodation for people affected tonight."
UK Cladding Action Group said: "Our thoughts are with those affected by this awful fire in Barking."
