Town turned the table on high-flying Barnsley for just a third League One win in 15 games and halt a run of consecutive defeats.
First half goals from Greg Docherty, after just 90 seconds, and Ollie Norburn put Town in a commanding position and Luke Waterfall's second half sealed it.
Askey has come under increased pressure from supporters after struggling to pick up wins in the league. But he made four changes for the Tykes' visit - including dropping skipper Mat Sadler - and the change to a 3-5-2 formation paid off.
Now the boss is calling for the Salop board to give him time to progress. He said: "You do feel the pressure, obviously you do. You know what the game's like nowadays.
"It's all about the result and everything has to be instant.
"You're also trying to work under those pressures. When you first come into a football club it's difficult.
"But hopefully I can stay in the job long enough to do it properly and Shrewsbury will see the results from it."
The boss continued: "I've been surprised by how supportive the supporters are. They really get behind the players and it makes a difference.
"It's a really good club and that's why I want to do well for the club - hopefully I can.
"But if I don't then it won't be for the lack of trying.
"It's a great club and everybody wants for it to do well."
Askey felt that the display was his side's best performance of the season.
He said: "It was probably our best performance. Just for the attitude really, work rate was what we were after, closing down, and that gives us a platform to play as well as we did.
"We came in at half-time and probably should have scored more goals.
"It's what I needed and what the club needed. You can't ask any more."
The Town boss dropped Sadler and made scorer Waterfall captain. On that he added: "The system we played suits the ones we played more than perhaps Mat.
"If we'd have lost tonight I'd have probably got slated because Mat's been doing well. It was a difficult decision but you've got to do what you think's right."