Historic headstones were knocked over and a 100-year-old stone fence smashed, leaving the cemetery at Govan Old Parish Church needing repairs costing £10,000.
Local community police officers traced five youths, aged between 10 and 14, who have been reported to the Children's Panel in connection with the damage caused on Tuesday and Wednesday night.
Two 10-year-old boys and an 11-year-old boy have been reported for malicious mischief in connection with an alleged incident on Tuesday.
A 14-year-old and an 11-year-old were reported in connection with vandalism the following evening to two 15th century headstones.
Locals say the incidents are just the latest in a two-month "reign of terror".
The latest alleged attacks were met with shock from the residents and members of the congregation, with one describing it as "just plain badness".
Another local source, said: "Some members of the congregation have been left distressed by the damage caused.
"The whole back section of the railings and coping stones have been broken and pushed over the edge.
"The staircase wall has been knocked down four times, the church broken into and now this fence line is being smashed a section at a time.
"Headstones, some of which were recently erected, have been knocked over too. Last month, there was even one kid spray-painting the graveyard wall in daylight."
Chief Inspector Simon Jeacocke, area commander for Govan and Craigton, said the problem would be tackled.
He said: "My message to anyone thinking about committing a crime is you are not going to get away with it.
"This kind of vandalism is despicable and I would ask those who are committing these crimes to stop and think about the consequences."
Increased police patrols have been stepped up in the area, in a bid to reassure residents.
Mr Jeacocke, added: "Officers are working with various partners, including Govan Church, to prevent damage and detect those responsible
"We have significantly increased the number of officers in the area, so there is an almost permanent presence, when possible
"High-visibility street briefings will be taking place at Govan Cross on Friday and Saturday evenings. This is to reassure members of the public that we are aware of the problems and taking action.
"I appreciate the support we have received from the local community and want to reassure them we are committed to tackling this issue. Anyone with information, or concerns, should not hesitate to contact officers."
The news that police have reported five boys comes after the Evening Times revealed a fall in charges made against under-18s in Glasgow.
New figures show that 7725 charges were made against under-18s in Glasgow in 2012-2013. That is a drop of 12% from 8734 a year before and down 45% from 13,981 in 2007-2008.
In 2012-13 there were 15,608 calls complaining about youths in Glasgow, down from 60,580 in 2006-2007, a drop of 74%.
The five boys will now go before a Children's Panel.