He graduated from Law Department of the Azerbaijan State University.
From 1954, he was the Senior Professor and then Dean of the Law Department of Azerbaijan State University, from 1957 he was the Dean of International Law Department. From 1965, he was the Director of the Department of Constitutional Law at the same university. In 1993–1996, he was the Rector of Baku State University.
Alasgarov was also an Honorary Doctor of the Kiev State University.
He was elected to the National Assembly of Azerbaijan in the 1995 parliamentary elections and re-elected in the 2000 parliamentary elections. He was then replaced by Ogtay Asadov in December 2005.
In the 2005 parliamentary elections, Alasgarov was re-elected to the parliament from Garadag district of Baku. He was also the Deputy Chairman of the New Azerbaijan Party.
According to media reports, Alasgarov"s health was deteriorating due to diabetes, which is why he did not run for re-election in November 2010.
On October 16, 1996 he was elected the Speaker of the National Assembly by the members of the parliament and re-elected on November 24, 2000.