Ousted president Zelaya insists voter turnout was low and calls vote illegitimate.
Lobo was declared the winner of the election on Monday and is due to take office in January.
He has called on neighbouring Latin American countries to support him.
However some regional countries including Brazil, Argentina and Venezuela have said they will refuse to recognise the result.
They say the election was illegitimate because it was held by a government installed in the wake of a military coup that ousted the democratically elected president, Manuel Zelaya.
Speaking to reporters on Monday, Luiz Inacio Lula da Silva, the Brazilian president, said that his nation would not rethink its rejection of the election because he said that might encourage coups in other nations.
Zelaya himself had called for a boycott of the election.
He remains holed up in the Brazilian embassy in Honduran capital, Tegucigalpa, having taken refuge there after sneaking back into the country two months ago.
Al Jazeera’s Latin America editor Lucia Newman, reporting from the Honduran capital, said major questions remains over what will happen to Zelaya in the wake of Sunday’s vote.
She said intensive negotiations had begun that were expected to last for several days as to what will happen to him, including whether he might be exiled or given an amnesty.
Zelaya himself has said he has no plans to move until what he calls the “dictatorship” installed following his removal is no longer running the country.
“We are fighting a dictatorship and until we defeat it we will not be satisfied,” Zelaya told Al Jazeera following the election.
Neither Zelaya nor the man who replaced him – Roberto Micheletti, the de facto president installed following the June coup – ran in the vote which was largely shunned by international monitors.
The election was organised before Zelaya was removed from power, with the candidates chosen in primaries last year.
Under a US-brokered pact, Honduran legislators will vote on Wednesday on whether Zelaya should be restored as head of that government, although that is not expected to pass.
Lobo wins vote rejected by Honduras’
Speaking to reporters on Monday, Arturo Valenzuela, the US assistant secretary for western hemisphere affairs, said Washington wants to see the unity government in place until Lobo takes office on January 27.
“Given the gravity of the coup d’etat and the polarisation that Honduras has undergone, both before and after the coup, it’s extremely important that Honduran leadership moving forward in the next few months [and] attempt to follow the overall broad frameworks of the Tegucigalpa-San Jose accord,” he said.
Lobo, the victor in Sunday’s election, has said he wants to put the coup and the issue of the ousted president behind him and move forward.
“Zelaya is just part of the past, it’s over”, he told reporters after attending a victory rally on Sunday night.
From his base in the Brazilian embassy, the ousted president has rejected the official election results.
Zelaya told reporters that information his supporters had gathered from polling stations indicated two-thirds of voters stayed home, which he insisted meant the election had no legitimacy.
According to the Honduran election commission however, almost 62 per cent of eligible voters cast their ballot, more than the 55 per cent it said turned out in the last election in 2005 which was won by Zelaya.
Few international observers travelled to Honduras for Sunday’s vote, but one group from the International Republican Institute, a US-based nonprofit organisation, said on Monday that they had witnessed “an election free of violence and overt acts of intimidation”.
In a statement, the institute called Sunday’s vote “credible and peaceful”.
The delegation was led by David Kramer, a former US assistant secretary of state, and included representatives from Mexico, the Netherlands, Peru, Poland, Spain and the US.