The South Bristol Link Road joins the A370 to the A38 and forms part of the route for the city's Metrobus scheme.
Simon Dunn from Alun Griffiths Contractors, which built the road, said: "It'll make it easier for the traffic to get to the airport."
Critics say it will worsen pollution and will fail to reduce traffic levels.
The cycle path section of the South Bristol Link Road is already open but safety checks are needed before the main road opens in January.
"The last thing we want to do is open the road just before Christmas and there are any concerns or any issues," said Mr Dunn
"The sensible thing to do is to leave it until January, do the safety checks and then open it."
Green Party prospective candidate for the Bristol South constituency Tony Dyer said: "Part of it goes through green belt and we're not convinced it's going to make a major difference to traffic.
"It will work for a small section of it but our other concern is how it's going to drive more demand for yet more roads."
The four councils behind the wider transport strategy also say the new road, which is 3.1 miles long (about 5km), will help boost south Bristol's economy, a view shared by the local Chamber of Commerce and firms in the area.
Chief executive of Computer Share UK, Naz Sarkar, said: "We have 1,500 people working here. Many of them live locally and they can travel by foot or cycle, they can use the free buses that we lay on for them.
"But we also have people who travel from much further afield, from Somerset, Gloucestershire, Wales and Wiltshire and for those people they do have to drive and the congestion in Bristol has made it difficult for them."
Hundreds of cyclists dressed in Santa outfits have been the first people to use a major new £45m road ahead of its official opening next month.