Hezbollah has rejected any move to force it to lay down its weapons, which it says are needed to deter Israeli attack. But its Lebanese opponents revived calls for the Shi'ite group to disarm after its military offensive in Beirut this month.
Tackling another of the challenges his presidency will face, Suleiman called for formal diplomatic links with Damascus.
Syria, Lebanon's main powerbroker for 29 years until 2005, has never agreed to exchange embassies with Beirut.
In New York, U.N. Secretary-General Ban Ki-moon said he hoped Suleiman's election presaged the revitalisation of all Lebanon's constitutional institutions and a return to dialogue.
Suleiman also urged dialogue, criticising a political discourse based on "accusations of treachery" which had "paved the way to divergence and discord, especially among youth".
Qatar's Emir Sheikh Hamad bin Khalifa al-Thani was the most prominent of many dignitaries in Beirut for the vote.
The majority and the opposition agreed long ago Suleiman should be president, but deadlock over the shape of a national unity government had forced the vote to be postponed 19 times.
The deal struck in Doha met the opposition's main demand for veto power in a unity government and secured the choice of a president on good terms with Syria and Hezbollah.
The agreement, which also stipulates a new law for 2009 parliamentary polls, has calmed a conflict that had stoked sectarian tensions, paralysed government and hurt the economy.
Parliament had not met for over 18 months, crippling Prime Minister Fouad Siniora's government. Bouts of violence killed scores and revived memories of the 1975-90 civil war.
Under Lebanon's complex power-sharing system, the president is always a Maronite Christian, the prime minister a Sunni Muslim and the speaker of parliament a Shi'ite Muslim.
Suleiman succeeds Emile Lahoud, an ally of Syria. Appointed army chief in 1998 when Damascus controlled Lebanon, Suleiman is inescapably linked to that era. He coordinated with Syrian troops before they withdrew from Lebanon in 2005 after an outcry sparked by the killing of former Prime Minister Rafik al-Hariri.
His first task as president is to appoint a new prime minister and consult with him on forming a cabinet, Siniora remaining as caretaker prime minister in the meantime.
Parliamentary majority leader Saad al-Hariri is frontrunner for the job, but his ally Siniora could stay on, officials said. Suleiman must nominate whoever is backed by a majority of MPs.
