The Scottish Football Association unveiled Burley as Alex McLeish's successor at a Hampden Park press conference on Thursday afternoon on a four-and-a-half year contract.
Burley, who previously managed Hearts, Derby County, Ipswich Town and Colchester, ended his two-year reign at St Mary's after being approached to take charge of his national team earlier in the week.
The former Scotland international won 11 caps for his country as a full-back and won the race to become the new national team boss ahead of Tommy Burns, Graeme Souness and Mark McGhee.
Meanwhile, Burley's departure has left Southampton searching for a new manager with several names already linked with the St Mary's hot-seat.
Former Saints boss Glenn Hoddle has been mentioned as a possible candidate, alongside ex-Derby boss Billy Davies and former Coventry City manager Micky Adams.
Gorman, who had been assistant to Burley and also Hoddle during his time at the South Coast club, plus first-team coach Dodd have been placed in charge for Saturday's FA Cup fourth round clash with Bury.
"This is a top job for someone and there are a lot of good candidates out there," chairman Leon Crouch told the Southampton website.
"We are disappointed to lose George Burley, but we will now sit down and look at all the options and decide how to move forward.
"In the meantime we are fully focused on two big home games in quick succession and we are confident the management team will hold things together strongly while we consider our options."