Moussa said the Arab League initiative had produced "a framework for understanding on sticking points.
"Therefore, solutions are there and the road is clear."
He urged rival factions to resume talks.
The present crisis began when six pro-Hezbollah cabinet ministers resigned last month after Fouad Siniora, the Lebanese prime minister, rejected their call for a government of national unity.
Hezbollah supporters have staged protests against Siniora's government, with ongoing sit-ins a few metres from Sinoria's Beirut office as part of their effort to force him to resign.
Hezbollah and its allies Amal and the Free Patriotic Movement (FPM) are currently demanding the aforementioned national unity government which would give them veto power over major government decisions.
"We have a president who acts alone on the basis that the government is illegitimate and we have a cabinet that claims in turn that the president has no legitimacy … It's a mess unprecedented in Lebanese history"
Maronite Christian patriarch
However, Siniora and his anti-Syrian supporters reject Hezbollah's demands, calling the campaign and protests since a Syria-backed coup.
While the Hezbollah-led opposition challenges Siniora's government, the latter disputes the legitimacy of Emile Lahoud, the Lebanese president.
Lahoud's mandate was extended until 2007 by a Syrian-inspired constitutional amendment in 2004, when Damascus had a solid presence in Lebanon.
Shortly after Moussa left Beirut for Cairo, hundreds of opposition supporters shouted "Siniora, get out" and waved Lebanese flags a few metres from the prime minister's office.
Nasrallah Sfeir, the Maronite Christian patriarch, said in a Christmas message: "The anarchy gripping state institutions is without precedent.
"We have a president who acts alone on the basis that the government is illegitimate and we have a cabinet that claims in turn that the president has no legitimacy ... . It's a mess unprecedented in Lebanese history."
Earlier, Moussa stressed that a lack of contacts between pro and anti-Syrian Lebanese factions was preventing a lasting solution to political and sectarian tensions in the country.
He said: "Contacts among various leaders are nonexistent, something that makes reaching an understanding very difficult."