A masked gang hijacked the vehicle at gunpoint in the early hours of Tuesday morning, before planting the bomb in the boot.
The driver was forced to travel with the device on board, from Cooke Street to Strand Road - where the bomb later exploded.
The PSNI received a telephone warning by a group claiming to be from Óglaigh na hÉireann, a dissident republican organisation, that the device would explode in 45 minutes but it went off just 23 minutes later.
Chief Superintendent Steven Martin said: "A taxi driver arrived at Strand police station and he told police that a few minutes earlier he had been hijacked and he believed there was a bomb in his car.
"Police immediately started an evacuation and a few minutes later, at 3am, police received a warning saying there was a bomb outside Strand road police station and it would go off in 45 minutes.
"Approximately 22-23 minutes after 3am and after that warning the bomb activated and went off ".
Police were still evacuating the area - which is close to a number of residential properties, including sheltered accommodation for the elderly - when the bomb exploded, at about 3.20am
A local business man told UTV he and his staff were lucky to be alive.
Logfi Jalloul, who owns a kebab shop opposite Strand road police station and saw the blast, said: "Five minutes later and we would be somewhere else."
"The car pulled up in front of the police station and I saw a man running inside," Mr Jalloul told UTV.
"I just thought it was a taxi driver waiting for a man to come out of the station.
"Fifteen minutes later, a policeman came inside (the shop) and said 'Get out, quick'. Five minutes later, I watched the car blow up."
"I'm still in shock and I think the shop has been completely destroyed," the businessman said.
"The main thing is nobody was hurt - we got out safe, myself and my staff."
The attack - which has been branded "murderous" by the Acting Chairman of the Policing Board, Brian Rea - has also been condemned by representatives from across political divides.
In a joint statement, the First and deputy First Ministers stressed that Northern Ireland would not be dragged back to the dark days of the past.
DUP leader Peter Robinson said: "As a community we must send out a strong signal that there is no place in society for those who would use the bomb and the bullet."
Sinn Féin's Martin McGuinness added: "The people clearly support the institutions and we will remain united in our resolve to work together for the greater good."
Mayor of Derry City, Colum Eastwood said: "This is an unbelievable attack on the people of this city, never mind on the Police Service of Northern Ireland. It is completely against the will of the people.
"If these people are really Irish Republicans like they claim to be, they should listen to the people of Ireland and move off the stage."
The explosion comes just two weeks after the city's successful City of Culture bid for 2013, but the SDLP's Mr Eastwood has said this is just a temporary setback.
"We won't let it hold us back," he said.
"There's always going to be people in this society who want to drag us back to the past.
"We're determined to move forward to the future with positivity, in the spirit of reconciliation and we're very determined to do that. These people will not stop us."
Dissident republicans have been blamed for what is the second attack on Strand PSNI station in recent months, following the firing of a mortar bomb at its walls at the end of May.
'Campaign of violence'
Describing the latest incident as "cowardly, dangerous and vulgar", Foyle SDLP MP Mark Durkan said it was fortunate no lives had been lost.
"Those responsible have achieved nothing and this campaign of violence will achieve nothing," he added.
Sinn Féin MLA and Policing Board member Martina Anderson called on those who support the bombers to "come forward and explain to the people of Derry how these bombs will in any way advance republican objectives."
She added: "Whichever grouping carried out this attack, they need to seriously reflect on what they are doing.
"Their actions are no part of a campaign to bring about Irish unity and they have little or no popular support."
Northern Ireland's Justice Minister, Alliance leader David Ford, said those behind the bomb attack had displayed a lack of regard for human life.
"I commend the bravery of those police officers who risked their lives in trying to warn those in the vicinity of the attack," he added.
"The way in which people came together to secure the City of Culture for Derry/Londonderry demonstrates the true spirit of the city; and I am confident that they will not be distracted from the important task of continuing to build a bright and vibrant future for the area."
But DUP MLA Willie Hay questioned whether police had sufficient resources to tackle the threat from dissidents.
"I do believe that there is a situation developing where ordinary people on the ground believe that dissident republicans are out of control," he added.
Security levels were also called into question by TUV leader Jim Allister, who said the latest attack came as no surprise.
"When terrorism is rewarded, as it has been in Northern Ireland, and in tandem security is dumbed down to pander to the ill-named 'peace process', it is sadly no surprise that fresh terrorism flourishes," he said.
"Last night's Provo-style bombers are no different, in either modus operandi or motive, from the Provo mentors.
"With security denuded and terrorism rewarded with the highest office in the land, I fear we can anticipate more terrorism which, make no mistake, suits the Sinn Féin strategy as on the back of it they demand ever swifter concessions as the false antidote to terror."
The proxy bomb was a tactic used by the IRA for a short time in the early 1990s, whereby people were forced to drive car bombs into military targets.
The latest report from the paramilitary watchdog had concluded that dissident republicans, while still active, had little support and were incapable of mounting a campaign like the Provisional IRA.
But the Independent Monitoring Commission did find that the threat posed remained "very serious".
Police have appealed for anyone with information about Tuesday morning's bomb attack to come forward.
Strand Road from Asylum Road to the Queens Quay junction, including Asylum Road to Queen Street, will remain closed for two days as officers continue their investigation.