At tea on Sunday's third day, England were 428 all out in reply to West Indies' 370 -- a lead of 58 runs.
It was far less of an advantage than England would have wanted on a good batting pitch but far more than looked likely when Strauss was out with his side still behind on 363 for seven.
West Indies, who bowled well, had to suffer a frustrating eighth-wicket partnership of 53 in 92 balls between Bresnan and Broad before the latter miscued a sweep against spinner Shane Shillingford and was caught for 25.
Part-time spinner Marlon Samuels, who scored a century when West Indies batted, polished off the innings with two wickets for one run in nine balls.
Bresnan was 39 not out.
Fast bowler Ravi Rampaul led the tourists' attack with three wickets for 75 runs in 32 balls.
The West Indies took four wickets in Sunday's first session before captain Darren Sammy dismissed Strauss for his second hundred in as many matches, following the opener's 122 in a five-wicket first Test win at Lord's that gave England a 1-0 lead in this three-match series.
Strauss was 102 not out and Kevin Pietersen 72 not out at the start of Sunday's play.
But Pietersen had added just eight when he was lbw to Rampaul's inswinger.
Despite his challenge, Pietersen was out for 80, ending a third-wicket stand of 144.
The West Indies took the new ball as soon as possible with England 299 for three off 80 overs.
And it brought rewards with Kemar Roach, who'd sent down eight no-balls Saturday, getting back into his stride with two wickets for six runs in 16 balls to reduce England to 308 for five.
Fourth delivery with the new ball, Roach had Ian Bell aiming across the line, lbw for 22 although the tourists had to challenge umpire Aleem Dar's original not out verdict.
Jonathan Bairstow, in his second Test, never looked comfortable against the short ball and, trying to turn a rising Roach delivery legside, got a leading edge to mid-on and was caught by Shivnarine Chanderpaul.
This was the first time in Strauss's 21 Test hundreds -- one shy of the England record of 22 held jointly by Walter Hammond, Colin Cowdrey and Geoffrey Boycott -- the left-hander had added more than six runs when not out overnight on a century.
But the 35-year-old found runs hard to come by Sunday, managing just 39 in 98 balls including four boundaries.
He was eventually out when, trying to drive, he was caught behind off Sammy. Strauss batted for more than seven hours in total, facing 303 balls with 22 boundaries.
