The Labour leader visited the region on Saturday and said more action was needed to fund flood defences in areas at risk.
Tweeting The Yorkshire Post's front page on Saturday, which bears the headline 'Not a national emergency', following Mr Johnson's claims, Mr Corbyn wrote: "Boris Johnson is wrong.
"This is a national emergency.
"Emergency funds should be released immediately to support families in need and all those affected by the flooding."
The politician met with residents in Conisborough, in Doncaster, after an earlier visit to the West Yorkshire seat of Pudsey.
During the visit to Leeds, he said: "Obviously we need much better flood management and prevention schemes," he said.
"It also means properly funding our fire and rescue services and properly funding our Environment Agency to deal with this.
"The Environment Agency has lost a fifth of its staff, the fire services have lost more than a fifth of their staff. They're struggling to cope with this."
Mr Johnson said the Government will "certainly stand by ready to help" after the country was hit by a deluge of water described as "almost biblical" by residents in Toll Bar, near Doncaster.
Many, including Mr Corbyn, have blamed the scale of flooding in the region on climate change bringing about more extreme weather conditions.
A woman who died after being swept away in floods in Derbyshire this week was named on Saturday as Annie Hall, the county's former High Sheriff.
Flood warnings remain in place for parts of South Yorkshire over the weekend.
The Environment Agency issued "danger to life" warnings for the River Don at Barnby Dun, Bently, Fishlake, Kirk Bramwith, Kirk Sandall, South Bramwith and Willow Bridge caravan site.
There are also 54 warnings telling people to take immediate action in light of expected flooding, along with 74 alerts urging residents to be prepared.
Train operator Northern advised passengers not to travel on some routes, including between Sheffield and Doncaster, Sheffield and Goole and Doncaster and Scunthorpe.