The Sons will be due compensative of the loss of the 34-year-old, who has agreed a two-year contract.
Gary Teale, who has been in charge of St Mirren since Tommy Craig's dismissal in December, was also on the shortlist.
St Mirren chairman Stewart Gilmour said: "He impressed with his ethos how he wanted to play football, how he wanted to take the club forward."
Murray was on a shortlist of five candidates who were interviewed this week.
Former Ross County manager Derek Adams and Brechin City's Ray McKinnon, who will be announced as Raith Rovers' new manager on Saturday, were among the candidates.
"He's ambitious, he's young, he's got his whole managerial career in front of him," said Gilmour of Murray.
"I thought all the candidates who applied for this job were excellent. We're getting a lot of good young managers coming through.
"Our target has got to be to get out of this league, but when that will happen, time will tell. We have to develop our young players."
Teale, who remains under contract as a player, will take charge of St Mirren for the final match of the season away to Hamilton Academical on Saturday.
"Gary's very disappointed," said Gilmour. "I think in the future he will be a superb manager.
"He has done a superb job for us, but a wee bit of experience somewhere other than St Mirren would have benefited Gary."