It is eight games since their last win.
Quinn said: "It was a difficult decision for me to make, particularly after the success of last season.
"But I had to put the interests of the club and its supporters first and felt that now was the right time to bring a fresh approach to the club."
Chairman Roland Wycherley added: "It is with tremendous regret that we are parting company with Jimmy, but unfortunately he was adamant that the time was right for change.
"Whilst the board were obviously concerned about the current League position we will be forever indebted to Jimmy for gaining us promotion back into the league last season, and wish him every success for the future."
First-team coach Chic Bates will take charge for tomorrow's game against Southend.
The Shrews have picked up just 11 points from their opening 14 games, two points from safety.