At the close of play on the second day, England were 259 for two in reply to the tourists' first innings 370, which featured Sammy's 106 – the all-rounder's maiden Test ton.
England were 111 runs behind, with Strauss 102 not out and Kevin Pietersen, who had struck a six and 10 fours, unbeaten on 72.
They had added an unbroken 136 for the third wicket as they both cashed in on an excellent batting pitch.
Strauss, who had gone 18 months without a Test hundred before scoring 122 in England's five-wicket series opening win at Lord's, posted his 21st century in 96 matches at this level today.
That meant the 37-year-old left-handed opener was now just one short of the England record of 22 Test tons held jointly by Walter Hammond, Colin Cowdrey and Geoffrey Boycott.
"It's always a special thing to get a Test match hundred," Strauss told the BBC. "Having not had one for a while, I'm not going to get blase about that now.
"It's nice to be back in form and I'm determined to make the most of it. It felt a lot easier today than at Lord's because I wasn't fighting against myself as much."
Alastair Cook, Strauss's opening partner, had twice been caught behind off Kemar Roach no-balls when, on 24, he edged paceman Ravi Rampaul to wicketkeeper Denesh Ramdin.
But Strauss and Jonathan Trott took England to 123 for one at tea with a brisk stand of 80.
But neither Trott nor England had added to their score when, on 35, he was lbw to a Rampaul delivery that cut back sharply.
Trott called for a review but the decision was upheld.
Pietersen wasted little time in attacking Shane Shillingford, left out at Lord's, by going down the pitch to loft the first ball he received from the off-spinner for six over long-on.
England had their moments of good fortune – Pietersen was lucky when, on 25, he saw an inside-edge off Rampaul just miss the stumps before going for four.
But the West Indies were discovering, as England's bowlers had before them, that there was little room for error on this pitch.
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