West Midlands Police said the animal was tasered before being placed in a wheelie-bin ‘to ensure public safety’.
The man and woman were taken to hospital for treatment for injuries which are not believed to be life-threatening.
The police say the dog, which is believed to be a Bully type breed, will now be put to sleep.
A Coventry Police spokesperson said: “We were called to Purcell Road in Coventry shortly before 4pm today (Thursday) to reports that a man and a woman had suffered injuries after being attacked by a dog.
"The man and woman were taken to hospital with injuries which are not life threatening, but are considered serious.
"The incident took place inside a flat but the dog then escaped into a communal area of the building.
"In order to bring the incident to a safe conclusion, officers tasered the dog, which is believed to be an XL Bully and then contained it carefully in a nearby wheelie bin to ensure the public’s safety and so the victims could be accessed and receive treatment. Those injured were known to the owner.
"Our enquiries are continuing and no arrests have been made at this stage.
"We understand this was a very distressing incident for anyone who witnessed it and as a force we are taking serious action against dangerous dogs and their owners.
"We will also be having extra officers in the area to offer reassurance. A decision has been made by the owner to put the dog to sleep.
"Witnesses or people with information should contact us via Live Chat on our website or by calling 101 quoting crime investigation number 20/849096/23."