Anderson, who became England’s leading Test wicket-taker in Antigua last week, was in effervescent form as he played a decisive role in six consecutive wickets to inspire his side on a flat pitch.
England were left to chase 143, with Cook (59no) and Gary Ballance (81no) both scoring their second half-centuries of the match to hand the tourists a 1-0 lead in the three-match series.
Cook told Sky Sports 2: “The new ball was really crucial. It was all about trying to put the ball in the right areas and credit Jimmy, what an outstanding spell that was.
“I can’t fault the lads – it was a tough wicket to get any kind of result, slow and flat. To get 20 wickets the way we did was a great effort, down to attitude, character and a lot of skill.
“You can always improve on a performance but the lads were brilliant. The wickets have been slow and hard to force a result on but we’ve put in a lot of hard work and got our reward today.”
Anderson said: “We stuck to the task brilliantly and never took our foot off the gas.
“I’m delighted we managed to get the wickets and get a result out of a game when we were looking at a draw on a flat wicket. We all believed we could get them and put the pressure on.
“Drawing games isn’t why we played the game – we want to win and this was a great team performance.”
Man of the match Joe Root, who scored 182 in England’s first innings, said: “It feels better now having the win behind us. The way we’ve applied ourselves over five days has been excellent, it’s something great to take forward in the series and for the rest of the summer.
“There are so many positives for us all to take and it’s fantastic – I can’t wait to get back out there in Barbados.”
West Indies captain Denesh Ramdin was left bemoaning the Anderson-inspired collapse which opened the door for an England victory.
He said: “We fought hard for the first four days but losing six wickets in that session cost us the Test. You should take the initiative but James Anderson took it away from us.
“After four hard days of cricket to lose the match in one session is really hard. But that’s how it goes – England went out and executed their plans and they had the conditions in their favour as the game went on.”
However, Ramdin was optimistic about the fitness of Jason Holder, who had left the field on a stretcher after rolling his left ankle mid-delivery before tea.
The third Test starts in Barbados on Friday and Ramdin said: “It’s just a slight injury. Hopefully he can rest up for the next few days and come back stronger.”