The hosts began the final day 10 for two in their pursuit of 191, but Cook (79) and Bell (63 not out) secured victory after Jonathan Trott and Kevin Pietersen had departed in the first hour of the morning session.
Strauss told Sky Sports: "I think the wicket was pretty slow and flat and we had a pretty good chance of chasing it down. I thought Alastair Cook and Ian Bell went about in a very civilised and professional manner and thankfully got us over the line. It was very sensible batting."
The England skipper was also quick to hail the efforts of the tourists, who put up a much more spirited fight than some critics had anticipated.
Dogged left-hander Shivnarine Chanderpaul struck 87 not out and 91, while his partnership of 157 with Marlon Samuels in the Windies' second innings frustrated England's bowling attack on Sunday.
"It's been a tough Test match, we've had to work very hard for this," said Strauss.
"West Indies have played some really good cricket. The way Chanderpaul and Samuels batted yesterday made life very difficult for us and we had to dig pretty deep. The seamers had to go through a lot of overs to take those wickets but when you have to dig deep it makes it a bit satisfying when you get over the line."
It was also a Test to remember for Strauss, who made his first century in 26 attempts - a run stretching back to November 2010.
"It was lovely to get that hundred and when you score a hundred in a winning cause, it makes it even more special but there were a lot of good performances in the side so overall we're pretty happy," added the left-hander.
"There are always areas we can try and improve and we'll try and do that in the second Test."