The operation, carried out under time pressure and "the watchful eye of the world", required 13 tug boats, said Boskalis CEO Peter Berdowski.
Egypt said it has ended the crisis.
The ship is now reportedly en route to the Great Bitter Lake, a wide stretch of water halfway between the north and south end of the canal, where it would undergo technical examination, according to canal authorities.
The stern of the ship was dislodged early on Monday, sparking immediate praise from Egypt's President Abdel Fattah al-Sisi, even as Boskalis initially warned that the bow would prove more hard to unwedge.
The giant container ship that blocked the Suez Canal for nearly a week has been fully floated and traffic in the waterway would resume, the canal authority said in a statement.
The impact of the canal shutdown has raised new questions about global trade practices, including the reliance on massive cargo ships, the conditions of workers on the vessels, and environmental degradation.
Although the vessel is vulnerable to damage in its current position, Shoei Kisen Kaisha Ltd., the company that owns the Ever Given, dismissed concerns on Monday, saying that the ship's engine was functional and it could pursue its trip normally when freed. While the Ever Given was stuck, the rising and falling tides put stress on the vessel, which is 400 meters (a quarter mile) long, raising concerns it could crack or break.
Suez ship partly refloated in first step toward reopening canal
About 15% of world shipping traffic transits the Suez Canal, which is an important source of foreign currency revenue for Egypt. Dave Hinton, the owner of a timber company in northwest England, said he had a consignment of French oak stuck on a ship.
"We've just searched the bottom and soil of the Suez Canal and thankfully it is sound and has no issues, and ships will pass through it today", he said.
While remaining volatile amid concerns about when the backlog of ships in the waterway that handles about 15 percent of global shipping would be cleared, at 10.46 (GMT) Brent crude oil futures for June delivery recovered 0.98 percent to $65.06.
The SCA has said it can accelerate convoys through the canal once the Ever Given is freed.
Crews have been working hard to refloat the container ship over the last few days.
A father and his family climbed to the roof of their red brick house to get a better view as the ship - laden with at least nine levels of large containers - slowly passed by.
Even as salvage work continued, President Abdel Fattah el-Sissi, who for days was silent about the crisis, praised Monday's events.
Egyptian authorities were desperate to get traffic flowing again through the waterway that's a conduit for about 12 per cent of world trade and about 1 million barrels of oil a day.
Firms including A.P. Moller-Maersk A/S and Hapag-Lloyd AG were forced to reroute their ships via the southern tip of Africa, which can add two weeks on to a journey between Europe and Asia.