Indications are that presidential results may be released on Thursday.
A two-thirds majority in parliament would allow Zanu-PF to amend Zimbabwe's constitution.
The president, who has called on the opposition to "lose gracefully", said worldwide election observers had told him they are "disappointed with the events of yesterday" that left at least six people dead.
Monday's polls the first since autocratic president Mugabe was forced out by a brief military takeover in November had been meant to turn the page on years of violence-marred elections and brutal repression of dissent.
Despite promises by the Zimbabwe Electoral Commission that they would soon state when exactly they would release the results of the presidential election, opposition supporters have said they would continue to protest.
"On many occasions, preparation, financing, media and hopefully not in the counting - it was advantageous for the ruling party", he told AFP as the mission called for transparency in the release of results.
United Nations Secretary-General Antonio Guterres called on Zimbabwe's political leaders and people to exercise restraint and reject any form of violence.
So far, the ZEC has said that of 210 parliamentary seats, 205 had been counted with ZANU-PF winning 144 and the MDC Alliance 61. Another 58 seats are yet to be declared.
MDC leader Nelson Chamisa, 40, has said the results were fraudulent.
Many Zimbabweans looked to the vote as a launching pad for a return to the kind of worldwide acceptance and relative prosperity that the country enjoyed in the first part of the rule of Mugabe, who took office after independence from white minority rule in 1980.
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"We have been in communication with Nelson Chamisa to discuss how to immediately defuse the situation, and we must maintain this dialogue in order to protect the peace we hold dear", Mnangagwa tweeted.
The European Union, which has been monitoring the elections for the first time since 2002, will issue its verdict later on Wednesday.
Zimbabwe was once one of Africa's most promising economies but under Mugabe's rule became tainted by corruption, mismanagement and diplomatic isolation.
Its population of 13-million is struggling amid shortages of foreign currency, unemployment above 80% and lack of foreign investment.
Many shops were closed on a quiet Thursday morning in Harare, where scattered debris, charred remains from fires and a few dozen soldiers patrolling the streets acted as a reminder of the violence that erupted a day before.
The commission, which must announce the presidential results by Saturday, has said the vote was free and fair.
Chamisa's MDC won in most urban centres, where it enjoys majority support.
However, Douglas Mwonzora, a top MDC Alliance official, told the BBC's Andrew Harding that the endorsement on Sunday of their candidate by Mr Mugabe had cost the party votes.
Zimbabwe's president said on Thursday (Friday NZ Time) his government has been in touch with the main opposition leader in an attempt to ease tensions after deadly election-related violence, a day after he accused the opposition of inciting it.
Anti-riot police backed by water cannon trucks monitored the demonstrators, while MDC supporters also protested outside the conference center where election results are being announced.