Strauss's men go into the second leg of a three-match series, which gets underway at Trent Bridge here on Friday, with a 1-0 lead after a five-wicket win at Lord's.
But the West Indies took England, the world's leading Test side, into the fifth day at Lord's and twice made them stumble in pursuit of a victory target of 191.
"It was very hard work (at Lord's), that's always the case with Test match cricket," Strauss told Sky Sports at Trent Bridge here on Thursday.
"Often it's hard work to get on top of the opposition and often it's hard work to nail that final nail in coffin.
"The West Indies put up a really good fight, there's times when they made life difficult for us and we were relieved to a certain extent to get over the line and win that game, but we're also aware we have to do it again this week if we want to win," added opening batsman Strauss, who ended his 18-month wait for a Test century at Lord's.
Alastair Cook (79) and Ian Bell (63 not out) saw England to victory at Lord's after they were in trouble at 57 for four, while the first Test was a personal triumph for seamer Stuart Broad, who took 11 wickets in the match.
"That target was tricky and there was a time when we were under a bit of pressure and Alastair Cook and Ian Bell dealt with that very well, and that's very important," said Strauss.
"There were some really good performances from our side.
"Obviously Stuart Broad bowled exceptionally well, most of the batters got a score in the game, so I think we're feeling in pretty good form heading into this match.
"But as I said before, we don't want to get ahead of ourselves, it's going to be very hard work this week, and matches here are always hard fought."
While England ponder whether to alter their seam attack, after all-rounder Tim Bresnan managed just one for 144 at Lord's, the West Indies will be forced into at least one change.
Fast bowler Shannon Gabriel, who took four wickets on debut at Lord's, was subsequently ruled out of the rest of the series with a back injury.
His absence could see the return of fellow quick Ravi Rampaul, who would have played at Lord's but for a neck problem.
"Last year-and-a-half Ravi has been the guy who always takes wickets with the new ball for us," said West Indies captain Darren Sammy.
"The way Kemar Roach is bowling and the inclusion of Ravi, that could be a very good combination for us."
And the tourists -- who in 22 matches against Nottinghamshire and England dating back to 1906 have never lost a first-class fixture at Trent Bridge, a run including four wins and four draws in eight Tests -- could recall off-spinner Shane Shillingford.
The Dominican was left out at Lord's, despite taking 10 wickets in his previous Test against Australia on home soil in Roseau last month.
But Sammy said he could return as West Indies look to improve on a record of just two wins in their last 31 Tests.
"Most definitely, we'll consider playing Shillingford," Sammy said.
"Looking at our attack, playing on flat pitches over the last year or so, a spinner has always been in our line-up, and we've gone with the two quick men and myself (a medium-pacer), and that has been successful for us.
"So we'll definitely consider that and hopefully we can go out there and take 20 wickets against England.
