The mile-long Northern Inner Distributor Road (NIDR) now links the east of the town with the west.
It was estimated in January that technical issues involving a bridge increased the cost of the £21m project by a further £10m.
The council is in a legal dispute with the contractor Carillion about the extra costs and who should pay.
Somerset County Council said the delays had been "enormously frustrating".
Cabinet member for highways, John Woodman, said the new road will "unlock" the old cattle market Firepool site for development and ease traffic in the town.
He added: "The dispute is ongoing and we will work hard to make the best result for all the people of Somerset and the people of Taunton."
Mike Rigby, independent councillor for Bishops Lydeard, said the council had "a lot of questions" to answer about how the project was managed.
"How has it been that the council signed a contract that enabled a contractor to take more than twice the amount of time to build what should be a fairly straight-forward job?
"It worries me, it's [an] example of how the county council has not well-managed a large contract."
Carillion had nothing to add to the council statement.

Colonel Steve Warren said more than 20 strikes have been carried out in the past four days, destroying IS positions.
The raids are in support of Iraqi forces who are trying to retake Falluja from IS.
Meanwhile there are dire warnings about conditions in the city.
The IS commander in the city, Maher Al-Bilawi, was killed in bombardments targeting IS fighter positions and gun emplacements, Col Warren said.
However, the colonel warned that it was unclear how long the battle to retake the city would last.
Islamic State: The full story
Some 50,000 civilians remain trapped in the city and have been told via leaflet drops to avoid IS areas and put white sheets on their roofs, Col Warren said.
The UN says it has reports of people dying of starvation and being killed for refusing to fight for IS.
"We have dramatic reports of the increase of the number of executions of men and older boys, refusing to fight on behalf of Isil," said Melissa Fleming, a spokeswoman for the UN High Commissioner for Refugees, using an alternative acronym for IS.
"Other reports say a number of people attempting to depart have been executed, or whipped. One man's leg was amputated reportedly."
The UN says hundreds of families were able to flee the city on Friday with the help of government forces.
The assault is being carried out by a combined force comprising thousands of Iraqi army troops, the federal police and Sunni tribal fighters, Col Warren said.
Militia forces were also involved but had pledged to remain outside the city, he added.
Falluja fell to IS in 2014, a key moment in its rise that saw it declare a caliphate across swathes of Iraq and Syria.
A multi-million pound road scheme to ease congestion in Taunton has opened more than two years behind schedule.
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The Islamic State group's commander in the besieged Iraqi city of Falluja is among 70 militants killed in coalition air strikes, the US military says.