Sheffield City Council contractors began removing eight trees in Rustlings Road at about 05:00 GMT on Thursday.
An independent report found no "arboricultural reason" to remove seven of the trees.
A councillor said removing the trees would save £50,000 needed to repair pavements which he claimed they were damaging.
Bryan Lodge, the council's cabinet member for environment, said: "I think this is the right decision to tackle these trees now and not leave it for the future."
The felling by the council's contractor Amey was carried out early in the morning on the advice of police and to "keep it as low-key as we could", he said.
The £50,000 estimated cost of carrying out remedial work to tree-damaged pavements and kerbs in the street was "not viable", he added.
The council said the work needed to be done as part of the £2bn Streets Ahead scheme to get rid of diseased, damaging or dangerous trees.
Residents in Rustling Road said the trees were healthy and should remain.
The Independent Tree Panel's report on Rustling Road was written in July and posted online about the time work started to remove the trees.
It said seven of the trees were "in good condition with good life expectancy" although some were causing damage to kerbs and pavements.
Three protestors were arrested on suspicion of preventing lawful work but were later released without charge.
Cllr Hodge said he would "rather people had walked away" than be arrested over the tree-felling, although he admired the protestors' passion.
Jenny Hockey, a retired sociology lecturer, was one of those arrested and said she was held in a cell for eight hours.
She was fast asleep at her house in Rustling Road when somebody "hammered on the door" and she saw two police officers.
She said: "You assume something horrific has happened.
"Out in the road it was the full works, police everywhere."
In April a High Court judge dismissed a bid for a judicial review into the tree-felling programme.
Sheffield City Council has felled more than 3,300 trees since 2012.
A dawn tree-felling operation that provoked outrage will save £50,000, a council has claimed.