Dozens of teenagers blocked a road with burning pallets and tyres during the incident on Thursday night.
The windscreen of a police vehicle was smashed and the fire service was stopped from entering the area.
It happened when police were searching a house at Altcar Park.
An imitation gun and a small amount of suspected cannabis were seized during the search and a 57-year-old woman and 33-year-old man were arrested.
They have been released on police bail pending further enquiries.
Galliagh resident Ailise McCallion said one youth told her they had burnt the shed in her garden and were going to target her oil tank next.
She said she now wanted to leave the area.
"I was terrified, I didn't want to aggravate them or say anything so I just went inside," she added.
"The whole night they were jumping in and out of my garden.
"They knocked stones off my wall, the lampposts they knocked them out, it was crazy."
Three windows at St Paul's Primary School in Moss Park were damaged as a result of the disturbances.
Its principal, Catriona McFeely, said she was disappointed at what had happened and warned someone could get "badly hurt".
"That is going to come out of our school budget, there is no other way of paying for that," she said.
"This is taking away resources from children in the school who are our main priority.
"We have already looked at security lighting for the outside of the school now we are going to look at security cameras."
Commenting on the trouble, Tommy Mullan, who lives in Galliagh, said some parents should be doing more to monitor their children.
"In this day and age, this is unbelievable," he said.
"What is wrong is there are too many young people running around here and you ask 'Where are the parents?'
"That is what it is all about, because if the parents were looking after them they wouldn't be out doing what they are doing."
Police have said up to 100 young people were involved in trouble in the Galliagh area of Londonderry, which saw officers attacked with bottles, stones and paint bombs.