Captain Andrew Strauss admitted England were forced to "dig deep" before sealing a in the first Test at Lord's.
They recovered from 59-4 early on the fifth day to reach their target of 191 by mid-afternoon.
"We had to dig deep but it makes it even more satisfying when you get over the line," said Strauss.
England later named the same 13-man squad for the second Test at Trent Bridge, which starts on Friday.
Alastair Cook and Ian Bell's 132-run stand for the fifth wicket carried England to the brink of victory after they lost Jonathan Trott and Kevin Pietersen inside the first hour on Monday.
"We certainly didn't have it all our own way," added Strauss. "West Indies made it difficult for us but we are delighted to be 1-0 up.
"We came here with expectations of winning and hopefully we can go on and do the same at Trent Bridge [in the second Test]."
Strauss answered his critics in the first innings with his first Test century in 18 months to put England in a dominant position.
"It was lovely to score that hundred and when you do that in a winning cause it makes it all the more special," he added.
"It has been a little while coming, there's been a bit of chat and it's good to get the monkey off my back. You know it's a talking point and the only way to keep people quiet is to go out and perform.
"I was happy and relieved to get those runs, but I never felt they were unobtainable."
Strauss paid tribute to man of the match Stuart Broad, who became the first Englishman since Ian Botham in 1978 to take 10 wickets or more at Lord's.
He added: "He got in a good rhythm; he bowled very well. In the second innings he had to work very hard for his wickets, but so did all of our bowlers."
West Indies skipper Darren Sammy insisted his team could take plenty of positives from their display heading to Nottingham.
"We did a lot of good things and we need to look at them and repeat them more consistently," he said.
"We fancied our chances in the morning but once the hardness of the ball went away, it became much easier to bat. An experienced batting line-up like England's showed their class and saw them home."
Sammy also singled out Shivnarine Chanderpaul for praise after he scored 178 runs over the two innings.
"That's something we are used to from him," he added. "He is the most experienced guy in the dressing room.
"I think he loves playing in England. The work he has been doing for the team has been very important and hopefully more of our batsmen can do the same."