Jonathan Trott and Kevin Pietersen both fell early at the start of the fifth and final day, giving the tourists renewed hope of causing an upset, with a further 134 needed for Strauss’ side to claim victory.
But Cook and Bell put on an outstanding fifth-wicket partnership of 132 which only came to an end when the former edged his effort to Kirk Edwards, but his 79 runs had left England just two runs shy of their target.
Debutant Jonny Bairstow entered the fray, but fittingly, it was Bell who sealed victory with a clinical four as England finished on 193-5.
“We felt the wicket was pretty slow and flat,” Strauss told the BBC.
“It looked a bit dicey at 40 odd for four but Alastair Cook and Ian Bell set about it in a very civilised manner and saw us home.
“It’s been a tough Test match. We had to dig deep but when you have to dig deep it makes it even more satisfying when you get over the line.
“It was lovely to score that hundred and when you do that in a winning cause it makes it all the more special.”
Bowler Stuart Broad, who took 11 wickets in the first Test, believes winning the toss helped England execute their game-plan and caused the tourists plenty of problems on day one.
“Winning the toss and bowling is a bit of pressure on the bowling group, but we managed to take nine wickets on the first day which set the game up nicely. We had to work very hard for this victory, so it feels very sweet right now,” added Broad.
England take a 1-0 lead into the second Test at Trent Bridge which starts this Friday.
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