George Chicken, 76, died at Rose Court Lodge in Mansfield, Nottinghamshire in November 2012.
Embrace All Ltd pleaded guilty to failing to ensure residents were not exposed to risks to health and safety.
Manager Amanda Dean of Ambergate, Derbyshire, pleaded guilty to failing to take reasonable care of persons affected by her work.
Nottingham Crown Court heard Mr Chicken, who had severe dementia, wandered through a first floor fire escape door in November 2012.
Mr Chicken, who had undergone hip and knee replacements, was confused, anxious and unsteady.
The court was also told he was known to wander around the home and had previously pushed open fire exit doors to get out.
His daughter Valerie Clowes said outside court: "To think all it would have taken to save my dad's life was a Â£5 thumb screw for the door he went thorough or a Â£20 child gate on his bedroom door."
He died in hospital 48 hours after his fall.
Two other charges against Dean were dropped.
The defendants will be sentenced in September.
A care home has admitted safety breaches after a resident died after falling from a fire escape staircase.