The chemical tanker and a Maltese-registered bulk carrier travelling on the same route collided at around 2am GMT on Tuesday morning, 90km west of the French port of Cherbourg.
The tanker was listing by 15 degrees four hours after the accident, according to French maritime authorities.
A British coastguard boat and a Royal Air Force helicopter evacuated the tanker's 22 crew members to a hospital on the British island of Guernsey, according to an RAF officer, who said there were no reports of serious injuries.
A British coastguard official, Fred Caygill, said that the tanker, loaded with 10,000 tonnes of phosphoric acid, was "significantly damaged" in the collision.
Caygill said the Marshall Islands-registered tanker was leaking phosphoric acid, which is a corrosive liquid that is soluble in water. However, he said the chemical did not pose an ocean pollution threat as its toxicity decreases greatly when dissolved in water.
'Text book' rescue
French authorities said there was no evidence of pollution so far, but that water pollution experts would assess the situation during the morning.
They said there were no reports of serious damage to the bulk carrier, the General Grot Rowecki, which had 21 crew on board and was carrying 26,000 tonnes of phosphate.
Choppy seas, with waves of up to 2.5 metres and winds of up to 40km an hour were reported at the time of the accident, but did not hamper the rescue effort.
Sergeant Tim Dickinson, of the RAF's rescue coordination centre, described it as "a text book rescue operation".
"We were very fortunate because the sea conditions were benign," he told the BBC.
The 198-metre cargo ship was headed for Poland's Baltic Sea port of Police, while the 126-metre tanker, the Ece, was bound for Ghent in Belgium.