They were called to Hill Top Mount, in Harehills, by the fire service at 1.30am last Wednesday (January 24) after the blaze at the mid-terrace property.
The mother and her son, aged five, and daughter, aged 14, had to be rescued from the house by firefighters and taken to hospital for treatment for the effects of smoke inhalation.
Specialist fire investigations later confirmed the fire had been started deliberately in the hallway through the letterbox and the trio had to flee to the upstairs of the house for refuge as they couldn't get out.
Now police want to trace the driver of the taxi, captured on CCTV, and say they could have "vital" information into what could have been "tragic consequences".
They say the taxi collected a fare from the roadside on the A58 Roundhay Road junction with Bayswater Grove next to BGT Landa Plumbing and Heating Supplies at about 1.35am that same morning and drove towards Sheepscar.
Det Insp Dave McDougal, of Leeds District CID, said: “The fire quickly spread through the downstairs of the house and to the stairwell leaving the victims no option but to seek refuge in the attic where they were rescued by firefighters.
“If it hadn’t been for the courageous actions of the fire service, this incident could easily have had very tragic consequences.
“We are treating this incident very seriously and are carrying out extensive enquiries, including checks on CCTV in the area.
“We are keen to trace the driver of a taxi that collected a fare from the roadside on the A58 Roundhay Road junction with Bayswater Grove next to BGT Landa Plumbing and Heating Supplies at about 1.35am on the Wednesday morning.
“The taxi, which is described as a black and white Hackney vehicle, continued city-bound down Roundhay Road towards Sheepscar.
“We are eager to speak to the driver of that taxi as we think they may have vital information that could assist the investigation. I want to stress that we are only interested in speaking to them as a potential witness and not for any other reason and I urge them to get in touch."
He also appealed for anyone else who witnessed anything suspicious around the time of the fire or has any information that could assist the investigation to contact Leeds District CID via 101 quoting Operation Arrowminster or call the independent charity Crimestoppers anonymously on 0800 555 111.