The visitors went in front with a well worked goal in the 29th minute when Tom Adeyemi raced on to a neat flick from Paul Hayes and rolled a low slot in to the bottom left hand corner of the net.
The Bees then doubled their lead nine minutes later when a low right wing cross from Sam Saunders was diverted past his own goalkeeper by Ryan Cresswell at the near post.
However, Southend got back in to the game within 60 seconds when Sean Clohessy's deep cross was the right flank was headed home at the far post by Barry Corr via the underside of the crossbar.
Corr then doubled his tally in the 53rd minute when he was left all alone in the penalty area after the Brentford defence charged out trying to play offside from Hurst's right wing free-kick.
"The players did very well to come back against a strong side who are doing well in the league above us," said Sturrock.
"For long parts of the game I'm not sure you could tell who was in what league and we created enough chances to have won the game.
"The players are gutted not to have got through but it was pleasing to see Barry get a couple of goals.
"He took them well and I didn't see a problem with the second one as one of their defenders was lazy coming out at the back post.
"There were players offside in there but I don't think Baz was one of them."