Two of the devices were used to attack the rear of Woodbourne PSNI Station in the Stewartstown Road area just after 10.30pm on Monday evening.
One exploded while the second failed to detonate. Police are blaming dissidents.
Army bomb squad officers were tasked to the scene but there were no reports of any injuries or damage, and police said the security alert has now ended.
Local people who were evacuated were allowed to return home on Tuesday afternoon following an extensive search of the area. The remnants of the exploded bomb were found.
Officers confirmed that the unexploded bomb was handed in by a group of youngsters.
They were prepared to expose local children to death and injury
Chief Superintendent George Clarke
A statement said: "Initial reports suggested the device handed in by the children was the exploded device, however we can now confirm they handed in the unexploded device."
An update from the PSNI said the bomb parts will now undergo tests.
It explained: "As a result of the search undertaken at Woodbourne PSNI Station today a number of component parts in relation to the exploded device have been found and these will now be subject to further examination."
Chief Superintendent George Clarke told UTV it was lucky no-one was killed.
He continued: "The clear responsibility for putting these children at risk rests full square with the people who recklessly, in their determination to kill the police officers that serve the community, exposed that community to greater risk.
"These devices were in no way stable, these are homemade explosive devices and we are extremely fortunate that both didn't explode and no-one was killed."
A local DUP councillor said the bomb was heard as far away as Finaghy and Lisburn.
Brian Kingston added: "A group aged around nine, 10, 11, mixed boys and girls, came round the side of the police station carrying an object.
"This is a combination of childhood innocence and a terrorist attempt to maim and kill. The thought of what could have happened to those children if that device had exploded - they could have been ripped apart - it doesn't bear thinking about."
Sue Ramsey of Sinn Féin condemned the attack.
The west Belfast MLA added: "This attack was not only irresponsible, it was reckless and put the lives of civilians, including children as well as PSNI personnel at risk.
"Those responsible need to explain what they hope to gain by these stupid and dangerous actions. There is no popular support for their activities in our communities and they should listen to the people, stop now before a child or somebody else is killed or seriously injured."
Ross Hussey of the UUP said: "Last night demonstrated better than words could ever say, just how dangerous a threat these fanatics pose to the entire community and how important it is that the community helps the police to remove this cancer from our midst."
Secretary of State Theresa Villiers said: "I condemn this despicable attack on a police station in west Belfast. Those responsible showed a complete disregard for life and contempt for local people. These terrorists will not succeed in dragging Northern Ireland back.
Police have linked the attack to dissident republicans.
Chief Superintendent George Clarke added: "Dissident republicans haven't moved forward and realised the way to advance any argument is political.
"No one has come forward to claim this (attack). I cannot see the logic of throwing bombs at the back of a police station and then leaving an unexploded one for children to pick up."
Policing Board Chair Anne Connolly said: "Thankfully those behind this attack failed in their aim to cause serious harm. Those involved showed complete disregard for the safety of the community and in this incident it is deeply concerning that the children who were in the area at the time were left exposed to such terror.
"I urge anyone with information to bring it to the police so these attacks can be stopped."