Adam Dugdale headed the only goal of the game in Saturday's clash at Gresty Road to hand Steve Davis' men a slender but potentially crucial advantage.
Former Southampton and Sheffield Wednesday boss Sturrock wants his players to be positive at Roots Hall on Wednesday night, but fears too much aggression could see Alex run away with the tie - citing the example of Blackpool, who turned a 1-0 first-leg advantage into a comfortable second-leg lead in their Championship semi-final with Birmingham last week.
"We're going to have to be very thoughtful defensively and make sure we don't leave the back door open," he said.
"I watched Blackpool the other day and they were 3-0 up very, very rapidly. You don't want to be too cavalier.
"We've also got to be positive in our approach - at the end of the day we have to score two goals.
"Away goals don't count, which makes it much easier. We know what we've got to do - either go 1-0 and win it on penalties or we get the goals that we need."
Sturrock struggled to hide his exasperation when discussing a performance that bore little resemblance to the team that took 13 points from their last five league games without conceding a goal.
Even Southend's better performers were not immune from criticism after a lacklustre first-half showing was compounded by Dugdale's 49th minute decider.
"Anthony Grant was solid but he still gave the ball away stupidly two or three times," the Scot added.
"We were sloppy, doing things that we don't normally do.
"To be fair to the players, they responded better in the second half and gave a better account of themselves, but that was miles away from how we've been playing lately.
Victory extended Crewe's unbeaten run to 17 matches - they last tasted defeat at Southend in February - but goalscorer Dugdale will not be satisfied unless that streak is improved further at Wembley in a fortnight's time.
"We want to go 19 games unbeaten - 17 is great but we've still got to go to their place and that's going to be very tough.
"They're a well-drilled side and they've only just missed out (on automatic promotion).
"We have the advantage, but the tie is still open. We're confident in what we do - we've got a lot of good players in there.
"We're confident, but we've got to make sure we keep our minds right because they'll be hurting.
"They're going to come out at us al