"Presidential power will always be a guarantor of the basic law and of the citizens' rights, [it] will continue to serve the people of Russia, the nation's sovereign interests."
Russian Orthodox Church Patriarch Alexiy II led the church service at the graveside followed by a parade by guards of honour.
A military band played a few bars of the national anthem introduced by Yeltsin, then switched to the Soviet melody that Putin reinstated as Russia's official anthem.
Dmitry Medvedev, who will take over as president, was also at the ceremony.
Critics of Putin accuse him of betraying Yeltsin's legacy by rolling back democratic freedoms, concentrating power in his hands and restoring attributes of the Soviet Union that Yeltsin helped overturn.
Yeltsin, who died aged 76, was a Communist leader who rebelled against his party's rule to become the first president of post-Soviet Russia.
His rule was marked by additional freedoms as well as economic turmoil and political instability that many Russians credit Putin with reversing.
Yeltsin stepped down from the presidency in December 1999 and handed over power to Putin, an ex-KGB spy.
Russia had experienced years of uninterrupted economic growth under Putin who has cited the "turmoil of the 1990s" as a contrast to his rule.
However, Putin has referred to Yeltsin with reverence.
The first anniversary of Yeltsin's death was marked as an event of national importance.
Yeltsin has had a street and a university named after him in his home city of Yekaterinburg in the Ural mountains.