Quinn, who is handing over the duties of chairman to former Home Secretary Dr John Reid, admits Strachan has a unique character which might "irritate". But he claims Celtic could not have a better boss, nor a better chief executive than Peter Lawwell.
"It's very important Gordon remains," said Quinn. "Gordon's style irritates some people - he's a wee bit idiosyncratic - but as a coach, there is no better coach in football than Gordon.
"He doesn't involve himself in the financial side or in transfers, he leaves that to Peter (Lawwell). It's a great division of responsibilities. The best two signings I've ever made? I'd say so."
Strachan was appointed in 2005 after Martin O'Neill stepped down to care for his unwell wife, and following the popular Northern Irishman has been a demanding task.
O'Neill's departure from Celtic was emotional, and Quinn said: "Gordon came in on the back of that, but he's winning the fans over and I'm sure if Celtic go forward Gordon will be as important a figure in Celtic's history as Martin ever was."