The filing of the petition also means Mnangagwa's inauguration, which had been scheduled for Sunday, will have to be cancelled until the court renders a decision on the matter.
Posting on his blog twitter handle, Chamisa said: "Our legal team successfully filed our court papers.We have a good case and cause!"
His new cabinet will come soon after his inauguration set for this coming Sunday.
Police were looking for Biti and eight other opposition leaders for allegedly fomenting violence following the disputed national election.
The MDC Alliance's legal representatives‚ Chris Mhike and Sylvester Hashiti‚ and his chief election agent‚ Jameson Timba‚ arrived at the court about 30 minutes before the courts closed for the end of the day at 4pm to file the papers.
Zimbabwe's electoral commission has said Mnangagwa won the election, the first without longtime President Robert Mugabe on the ballot form, garnering 50.8 percent of the vote against 44.3 percent for Chamisa. "We are seeking a declaration to the effect that the presidential election was not properly conducted", MDC party lawyer Thanbani Mpofu said.
Rains to intensify; Red Alert in 7 districts
The chopper of the chief minister could not land at Kattappana in Idukki due to bad weather. Four teams of the National Disaster Response Force left from Chennai to Kerala .
The inauguration should take place within 48 hours of the court's ruling, it added. "But if they want to meet us in court we have sufficient legal minds to meet them squarely in court", ZANU-PF legal secretary Paul Mangwana said. Six died after soldiers opened fire on the protesters in a response the opposition alliance called "disproportionate and unjustified". The opposition said the result was rigged.
Also on Friday, lawyers for senior opposition figure Tendai Biti asked judges to throw out charges against him over the protests against alleged election fraud, in a case raising further worldwide concern about the new government.
He faces charges of inciting the protests last week by proclaiming victory for the opposition.
USA ambassador to Zimbabwe Brian Nichols Thursday made a surprise appearance at Tendai Biti's court hearing telling journalists soon after that the decision by Zimbabwean authorities to charge the MDC Alliance politician was "worrisome".
The Zimbabwe Human Rights Commission, established under the 2013 constitution, on Friday released a damning report into the post-election crackdown.
The president-elect who is to be sworn into office this weekend and his party, Zanu-PF, have welcomed the court action.