Nearly 10,000 fans paying as little as B£1 under a 'pay what you can' promotion saw Marcello Trotta's first-half volley clinch the points against an adventurous Shrewsbury side.
The win, which moved Brentford up to fifth place, came a week after a small section of home supporters called for Rosler's head.
But the German said: "The fans at this club are passionate and that's what makes it such a special club.
"But they have to remember that the players, coaching staff and the manager are the same.
"The relationship between fans and club is a bit like a marriage.
"Clouds come, but it doesn't mean you get divorced. We are all passionate at this club and I wouldn't want it any other way."
Rosler said early-season injuries meant his side had failed to find their rhythm and balance until now.
"We are clicking into gear and moving a lot closer to our best performances of last year," Rosler added.
"We were solid at the back and very composed in all areas.
"But I have to pay tribute to Shrewsbury, who came here to try to win instead of just parking the bus as many sides have done this season."
Unfortunately for the Shrews their ambition was undone when Trotta was in the right place to smash home at the near post after a jinking run and cross from Kadeem Harris in the15th minute.
The Italian almost made it two just before the break when he was played in by Adam Forshaw, but his rising drive was tipped wide by Shrews goalkeeper Chris Weale.
John Taylor gave the visitors their first attempt on goal just before the break, but his low drive was easily saved by David Button.
The goal Brentford wanted to seal the points almost came in the dying minutes, but substitute Sam Saunders rolled a low cross wide after good build-up play from Clayton Donaldson.