At the new party's first-ever conference in Greenock, Mr Salmond – who has led the party on an interim basis since its inception in March of this year – was elected unopposed, receiving a standing ovation from the few hundred delegates at Greenock Town Hall.
Mr MacAskill, one of the most high-profile defections from the SNP to the new party, won the depute leadership by 837 votes to 323, defeating Glasgow councillor Michelle Ferns.
The East Lothian MP was among the first to shift from the SNP to Alba, which fielded candidates in this year's Scottish Parliament election but failed to win a seat.
Another high-profile defection, Chris McEleny, was also confirmed as the party's new general secretary – a post he has also held on an interim basis.
His father, Inverclyde councillor Jim McEleny who was chair of the SNP group on the council, was also revealed on Saturday to have joined the party, citing the SNP’s lack of urgency on independence.
In a speech which was a direct challenge to the SNP, also holding its conference, Mr Salmond said: "People want a party which puts independence first and foremost, which sees independence not as something for the hereafter, but for the here and now.
“Alba is that party, we're fulfilling that desire in Scottish society and the strength of our delegates and membership indicates we're on the right road.
“Alba has been around for five or six months, but you're not really a party until you've had your conference. Up until today, I was the only nominated official in the whole party.”
Mr MacAskill later told delegates that Alba was Scotland’s fourth largest party with over 6,000 members.
He added: “We didn’t get the result we wanted in May. Others have to answer why, when Scotland’s situation is so critical and the plight of many Scots so perilous, they preferred to see unionist MSPs returned, rather than those committed to independence.
“But we’re alive and we’re kicking. We’ve a party to build, elections to fight and a cause to deliver.
“It cannot await recovery as there can be no recovery without independence.”