A stunning performance with the new ball from the fast bowler broke Windies second innings resistance.
Anderson went on to add two catches and a brilliant run out as the hosts were broken inside a crazy two hours in the morning session.
Cook insisted that Anderson’s intervention was the obvious turning point in a match that England won by nine wickets to take a 1-0 series lead.
“We had turned up thinking we had a chance but it was going to be hard work but when you have special players like Jimmy who can do fantastic things it is always possible that the opposition will crack," said Cook.
“That is what happened. It has taken us eight or nine days to make that happen - these are tough wickets to force a result but it makes it a great feeling and one we will remember.
“Jimmy may be coming to the latter part of his career but the skills he has learnt over that career and the experience he has is invaluable."
Cook added: “Sometimes we take him for granted I am very happy to have him in my side and to have played with him. He is still capable of showing fantastic skill and heart.”
Anderson finished with four for 43 but it was a spell of three for 14 in seven overs with the new ball that broke the West Indies.
After bowling the hosts out for 307, England chased down a target of 143 for the loss of just one wicket, Jonathan Trott out for a duck.
Cook (59) and Gary Ballance (81) steered them to a first overseas victory since Kolkata in 2012 and a first victory here in the Caribbean since 2004.