His election ends a period of uncertainty as he replaces Chabur Goc Alei, who was suspended by the SSFA's general assembly last year.
Amin beat former international referee Lual Maluk Lual in the polls which were overseen by delegates from Fifa and the Confederation of African Football.
"Let's forget our differences but be united by the beautiful game of football," Amin said.
"I welcome all my opponents to come and join hands so that we develop football in our beloved country South Sudan."
Alei was forced out of office amid allegations of mismanagement.
Amin is a well-known businessman in Juba and is a former board member with local club Atlabara.
Venasio Deng Amun will be Amin's deputy after he beat Sestilio Juba Leriba in the polls for vice-president.

No batsman from Bapchild Cricket Club was able to get off the mark against Christ Church University in Canterbury.
"We couldn't believe it, all they needed to do was hit a wall to get one run," Christ Church player Mike Rose told the Crawley Observer.
Somerset club Langport set the record for the lowest score when they were dismissed for zero in 1913.
Wirral CC were bowled out for three in a Cheshire League Division Three fixture in 2014 - though they had 11 players and were 0-8 at one stage.
And Kent village team Saltwood CC scored 216 before bowling Martin Walters out for zero in 8.2 overs during a match in 1964.
In first-class cricket, the lowest score ever made is six - by "The B's" against England at the old Lord's ground in 1810 - while New Zealand's 26 against England in 1955 remains the lowest total scored in a Test match.

Read the team news for Saturday's matches below.
Kick-off 15:00 unless stated. All times GMT.
Manchester City v Chelsea (12:30)
Crystal Palace v Southampton
Stoke City v Burnley
Sunderland v Leicester City
Tottenham Hotspur v Swansea City
West Bromwich Albion v Watford
West Ham United v Arsenal (17:30)
Birmingham City v Barnsley
Blackburn Rovers v Huddersfield Town
Bristol City v Ipswich Town
Burton Albion v Rotherham United
Cardiff City v Brighton and Hove Albion
Fulham v Reading
Norwich City v Brentford
Sheffield Wednesday v Preston North End
Wigan Athletic v Derby County
Leeds United v Aston Villa (17:30)
Francis Amin Michael Paul has been elected as the new president of the South Sudan Football Association (SSFA).
[NEXT_CONCEPT]
A cricket team was bowled out for 0 in just 20 balls in a county six-a-side indoor championships match.
[NEXT_CONCEPT]
Four of the Premier League's top five are in action on Saturday, while there is a packed programme in the Championship.