Soviet Premier Alexei N. Kosygin, the second most powerful man in the Kremlin during the 16-year era of Leonid Brezhnev, stepped down from his government post today at his own request because of poor health.

The prime minister, who is 76 and suffered a severe heart attack last year from which he has not recovered, was immediately succeeded by Nikolai A. Tikhonov, 75, a Brezhnev protege who has been deputy premier for the past four years and a full Politburo member since 1979.

Kosygin's retirement was announced in measured tones by Brezhnev to a meeting of the Supreme Soviet, the figurehead parliament, in the Great Kremlin Palace this afternoon. It marked the most significant leadership change since Brezhnev and Kosygin came to power in October 1964 after Nikita Khrushchev was ousted as Soviet leader.

Brezhnev, party chief since then and president since 1977, said Kosygin also asked to be relieved of his seat in the ruling Politburo.The request is expected to be granted soon after a regular leadership meeting.

The choice of Tikhonov, a sallow, square-faced industrial specialist, seems to lay the groundwork for an immediate post-Brezhnev Kremlin leadership of elderly men who share his conservative views on political and economic centrism.

Brezhnev told the 1,500 parliamentary delegates he received a letter yesterday from Kosygin requesting that he be relieved of his premiership because of his health, "which has been worsening lately." The letter thanked the party and Supreme Soviet for their confidence in him. In view of the request, Brezhnev said, he was asking the parliament to grant the prime minister's request, and said the Politburo "suggested" Tikhonov be made the new premier. The Supreme Soviet unanimously approved both changes in separate votes.

Kosygin's career in the hierarchy began in the middle of the Stalin purges in the mid-1930s. His regisnation is the first time in the 63-year history of the Soviet state that a party chief or premier has retired honorably.

As head of the government's Council of Ministers, his responsibility embraced managing the entire economy. But Soviet output throughout his tenure has lagged behind party expectations. From 1976 to 1980, the overall economy has expanded by only two-thirds of the planned goals.

Kosygin backed economic reforms in the mid-1960s, but lost out to Brezhnev, who steadily consolidated power to emerge in the 1970s as undisputed leader. Brezhnev capped his success by shoving Nikolai Podgorny aside as president three years ago in the last major leadership change.

Kosygin dropped out of sight intermittently during his premiership, apparently because of heart problems. Unconfirmed reports have circulated in Moscow in recent years that he long has wanted to resign. But last June it was announced he would deliver a keynote economic address at the upcoming 26th Party Congress in February, when the 1981-85 five-year plan will be adopted.

The address now presumably will be given by Tikhonov, a metallurgical engineer who first became associated with Brezhnev in the Dniepropetrovsk industrial heartland of the Ukraine in the 1940s. Like other longtime Brezhnev associates, such as Andrei Kirilenko and Konstantin Chernenko, Tikhonov is considered part of the "Dnieper group," or "Dnieper mafia," whom the party chief has steadily advanced to his inner circle.

Tikhonov's rise to the center of power has been marked by some apparent difficulties, possibly because he is a Ukrainian among a leadership dominated by Great Russians. He met Brezhnev during the postwar reconstruction of Ukrainian industry, when he was director of a pipe and tube mill. He was brought onto the Central Committee 14 years ago with many other veteran Brezhnev acquaintances, and then headed the Soviet steel industry for many years. He was made a deputy prime minister under Kosygin in 1965.

But when he became first deputy premier four years ago, he was not elevated to the Politburo, a promotion customary for that post, and had to wait two years before gaining nonvoting (candidate) status at the edge of the inner circle. But when Kosygin was felled by his latest heart attack last autumn, Tikhonov was given full Politburo membership and for the past year has virtually run the Soviet economy.

Kosygin reappeared in public late this spring, but has not been seen since early summer.

Observers here were struck by the ungraceful manner in which Kosygin's resignation was treated -- no tributes, no special praise for a man whose health had failed him in the service of his country. If nothing else, this suggests the transition mechanism within the Kremlin is by no means smooth.