"I feel I can make the people I work with better and if I can do that we have a chance. It is an exciting division to come into."
Former England defender Southgate could not prevent Boro's relegation from the Premier League last season and was dismissed on 20 October, just hours after his team defeated Derby 2-0.
He has since claimed that Boro had interviewed Strachan about replacing him more than two weeks before his departure.
Strachan is a former Scotland international midfielder who played for Dundee, Aberdeen, Manchester United, Leeds and Coventry during a successful career.
He became player-manager of Coventry in 1996 and remained at the club until his dismissal at the end of the 2000-01 season.
Strachan took over at Saints in October 2001 and took the club to the 2003 FA Cup final before resigning in February 2004.
The Scot succeeded Martin O'Neill as Celtic boss and won the SPL title in three consecutive seasons, but left the club after failing to retain the title last season.
Preston 2-2 Middlesbrough
Strachan takes over at Boro with the club fourth in the Championship after a 2-2 draw with Preston on Saturday when caretaker manager Colin Cooper was in charge of the team.
"I know you get time here, and the stability of the club means you have the chance to develop things the way you would like to see them develop," said Strachan.
"Celtic was fantastic, and I hope it can be as enjoyable here."
Gibson has appointed rookie managers Bryan Robson, Steve McClaren and Southgate during his reign as chairman, but indicated he wanted a more experienced name in charge.
"When I sat down and considered our start to the season, I felt that the league table was actually more favourable than some of our performances," Gibson said after Southgate's exit.
"Then I looked at the important games against the teams immediately around us, and took those results into account.
"In general the results and the performances have not been to the level I would have hoped for."
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