The baby girl was found with high levels of the gas in her blood, along with parents Chris Humphries and Lucy Deavall and her older daughter.
The family had been suffering headaches and dizziness for several weeks before getting tested by a doctor after the two-year girl started vomiting.
They had only been back at home in Mesnes Green in Lichfield a day following the birth of the newborn baby.
A Health and Safety Executive investigation found registered gas engineer Mark Whitfield, aged 33, of Burntwood, had carried out a landlord's gas safety certificate examination at the house before the family moved in.
However, he failed to spot evident defects with the boiler and flue.
Whitfield returned to the property several times once the house had been let, but repeatedly failed to spot the problems.
Appearing at Stafford Magistrates' Court, Whitfield, of Queen Street, Burntwood, pleaded guilty to breaching health and safety laws this week.
He was handed a 12-month community order with a requirement to carry out 240 hours' unpaid work.
He was also ordered to pay £1,185 costs and £1,000 compensation to the family.
After the hearing, HSE inspector David Brassington said: "This was a vicious storm of circumstance.
"Since the baby had been born the family were using the boiler more than usual to heat the house, both for the baby and the amount of visitors they were getting.
"It was that increased use of the boiler, coupled with another onset of illness, which made them suspect they were being poisoned.
"Mark Whitfield's failures over a series of visits exposed the family to significant risk to their health whenever the boiler was in operation.
"As a qualified gas engineer he should have identified the faults, classified the boiler as 'at risk' and initiated remedial works."
The court heard how the first bend in the flue was on top of the boiler when it should be more than 500mm above the top of the boiler.
The end of the flue was also not correct and there were signs of condensation and staining inside the boiler, which were likely to have been visible when the gas safety check was conducted in November in 2012.
The family discovered the carbon monoxide poisoning six months later.
He added: "The actions of qualified gas engineers are paramount to the safety of gas users.
"There are approximately 12 fatalities a year arising from carbon monoxide poisoning from gas appliances that have not been properly installed, ventilated or maintained. Mark Whitfield is extremely fortunate that this family did not add to that number."