Mnangagwa's inauguration planned for Sunday was postponed until the court makes its ruling. This was according to Legal and Parliamentary Affairs Minister Ziyambi Ziyambi. "We have a good case and cause".
Jameson Timba, a senior member of the MDC, said his party "will rest when this country is liberated".
The legal challenge is likely to result in a postponement of Mnangagwa's inauguration ceremony as president of Zimbabwe.
Mnangagwa narrowly won the presidential race with 50.8 percent of the vote - just enough to avoid a run-off against the MDC's Nelson Chamisa, who scored 44.3 percent.
Judges could declare a victor, call another election, or order a run-off or recount, the Veritas legal group said.
The MDC reported that its foreign lawyers had initially been denied entry into the Constitutional Court on work permit grounds.
Worldwide monitors largely praised the conduct of the election itself, although European Union observers said that Mnangagwa, a former long-time Mugabe ally, benefitted from an "un-level playing field" and some voter intimidation.
"On the basis of the evidence we have placed before the court, we seek in the main relief to the effect that the court should declare the proper victor and the proper victor is my client", Mpofu told reporters outside.
"We are seeking a declaration to the effect that the presidential election was not properly conducted".
The courts may take one day or three days but that has to be within the 14 days.
"I am ready to protect the millions who voted for my client".
The move is in a bid to strengthen the revolutionary party.
"If [the judges] think the evidence is weak they could throw the case out fairly fast".
Zanu-PF secretary for finance Patrick Chinamasa earlier in the week revealed that the party has assembled a 12-man legal team to respond to the challenge.
Two days after the vote, six people were killed in an army crackdown on protests against the victory by Mnangagwa's ruling ZANU-PF party, a reminder that Zimbabwe will struggle to fix its image as a nation known for repression and economic collapse.