Rosler was delighted his players maintained their discipline against a side he believes could have caused them problems if they had taken them lightly.
The German said: "I was very pleased with the energy and desire we showed and the way we pressed them in all areas of the field.
"Staines could have done to us what we did to Chelsea last season, but our professional approach to the game ensured we didn't slip up."
Rosler added: "We kept the tempo high and that kept Staines on the back foot.
"We didn't let them get a foothold in the game - all in all, a very solid performance. Most pleasing because we were determined to keep the clean sheet."
The game was as good as over before the break as Brentford cruised into a three-goal lead in the sides' first-ever Middlesex derby.
Former Brentford trainee Lewis Ferrell gave his former employers the perfect start when he handled in the box for Alan McCormack to make no mistake from the spot.
The Bees stretched their lead just after the half-hour when Jake Reeves rounded off a slick five-man move to slot home coolly under Swans keeper Kyle Merson.
Kadeem Harris then picked up the ball on halfway and ran on to rifle a 30-yard rising drive into the top corner.
Marcello Trotta made it four-nil just after the interval when he guided McCormack's precision right-wing cross over the keeper as Brentford turned the screw.
Staines' only real chance in a one-sided contest came after 68 minutes when Jerel Ifil bundled Max Worsfold's teasing free-kick wide when well placed.
Substitutes Will Grigg and Farid El Alagui both went close for the hosts before Clayton Donaldson rubbed salt in the sounds with a cheeky backheel to make it five-nil in injury time.
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