Sultanov said Aliyev is a Georgian citizen who grew up in the same village in Georgia as Farda Gadyrov, the gunman.
Gadyrov, 29, entered the academy and climbed up five floors, shooting everyone he met along the way, according to a joint statement from the interior ministry and state prosecutors.
He turned his pistol on himself when he saw police approaching, the statement said.
Anar Gadyrli, a health ministry spokesman, said 10 injured victims remained hospitalised, five of them in serious condition.
In an interview published in Azerbaijani media, Namiq Aliyev, a man identified as Aliyev's brother, said Nadir Aliyev worked at a fast-food shop not far from the oil academy and was acquainted with the gunman but had not been in contact with him recently.
Namiq Aliyev was quoted as saying his brother had been detained, questioned and released on Thursday, but then detained again.
Authorities have said nothing about a motive for the shooting.
ANS, an Azerbaijani television station, quoted an official in Dashtepe, the Georgian village where Gadyrov grew up, as saying that Gadyrov had left the village with his parents about a decade ago to live in Russia, then returned briefly about a month ago before moving to Azerbaijan.
The official, Vidadi Gasanov, described Gadyrov as an unsociable child, and said he had rarely left his home during his brief return.
Georgia borders Azerbaijan to the northwest and has a sizable ethnic-Azerbaijani minority.
Azerbaijan has experienced a post-Soviet boom fuelled by its rich Caspian Sea oil fields, and the oil academy has long been recognised as a major international centre for the training of oil industry specialists.