There was widespread speculation Strachan would be sacked after last Friday's 3-0 loss to England at Wembley.
Scotland were the only team from the British Isles not to qualify for Euro 2016 and are currently six points below group leaders England in qualifying for the 2018 World Cup.
There was widespread speculation Strachan would be sacked after last Friday's 3-0 loss to England at Wembley, but following a review, the SFA has decided to stand by him.
"We support Gordon unanimously in improving our qualification prospects," Regan said in a statement.
"The board have considered the team's performance since the start of the campaign and subsequently discussed the disappointing points total after four matches.
"Gordon shared the disappointment of the board and, of course, the fans. He is adamant, however, that we can recover the position and believes a play-off place (second in the group) still to be attainable.
"The board are convinced that he still has the hunger for this challenge and we have four home qualifiers in 2017 to rejuvenate our campaign."
While only the top team in each UEFA qualifying group will be guaranteed to qualify for the 2018 World Cup in Russia, the teams finishing second will go into the play-offs.
Scotland currently trail second-place Slovenia by four points.
"We were all extremely disappointed with the result on Friday night, but having reflected on our performance, I still believe we can get ourselves back into contention in the group," Strachan said.
"There are still 18 points to play for and the players, the staff and I, as in previous games, will give everything we have to turn things around."
Strachan, a former Scotland midfielder, succeeded Craig Levein in January 2013 and has a win percentage of 45.45 from his 33 games at the helm.