The rural grave, discovered last November in the town of Tetelcingo, consists of two 10m (33ft) deep pits.
Prosecutors say that the bodies may have been dumped illegally by morgue officials, but the investigation into who is responsible is ongoing.
Morelos is among the worst-affected states in Mexico's epidemic of drug-related violence.
At least 20,000 people have disappeared across Mexico, the UN estimates - other organisation put the number far higher.
Investigators at Tetelcingo worked under a yellow tent as families of missing persons and National Human Rights Commission representatives looked on.
Genetic samples will be taken from each set of remains to attempt identification before they are reburied in marked graves.
The exhumation is expected to take about five days.
State prosecutor Javier Perez Duron said an investigation was under way into who was responsible for the illegal mass burial.
"It is extremely important and the highest responsibility to count the bodies and give them a dignified burial," Mr Duron said.
Experts from the Autonomous University of the state of Morelos (UAEM) have been called in to help with DNA testing.
Alejandro Vera, rector of UAEM, said: "We recognize the tireless struggle of the relatives and victims' organisations of the more than 30,000 missing in this country that's sinking into barbarism."
Mr Vera founded a program to help those whose loved ones have disappeared in Mexico's drug war.
At least 20,000 people are thought to have disappeared.

England will play their first day-night Test when they host West Indies at Edgbaston in August.
Yorkshire say they were asked by England and Wales Cricket Board (ECB) about their willingness to move one of the Championship games at Headingley and would be interested in doing so.
The club's chief executive Mark Arthur called it a "great opportunity".
He said: "Until the fixture programme is published, we don't know when it will be or who the opposition is likely to be. But because we have got the best lights in the country for cricket purposes, then it's likely that we would be one of the selected grounds to host such an event.
"Hopefully when the fixture list comes out, it's something that all members and stakeholders will save a date for in their diary.
"Whether you're a club cricketer or a member who works 9-5 you'll be able to come along and watch two or three hours of Championship cricket after work and it will be very interesting to see what the uptake is on this initiative."
The start of the second ever day-night Test match between Pakistan and West Indies earlier in October drew a crowd of just 68 in Dubai.
Mexican authorities have begun exhuming 116 bodies found buried in a mass grave in the central state of Morelos.
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Yorkshire have said they would be interested in hosting a day-night game in the County Championship in 2017.