Officers fired flash grenades and rubber balls along with the tear gas as troublemakers wearing black masks and hoods confronted them in the street and pelted them with stones and other objects.
The confrontations broke out near the start of the main labor march near Montparnasse train station and resumed when police tried to disperse stragglers at the finish, near the Place d'Italie in southeast Paris.
The French Interior Ministry said 24 protesters and 14 police officers were injured. The ministry said 28,000 people marched in Paris and more than 164,000 in May Day rallies across the country.
A private company hired by a group of French news outlets, Occurrence, counted 40,000 protesters in Paris, while the CGT union said there were 80,000 participants.
More than 7,400 police officers from across France were deployed in Paris because of the May Day events. The Paris police department said there were 330 arrests. Officers also performed more than 15,300 "preventive searches" of bags.
The streets of the French capital calmed down on Wednesday evening as marchers slowly left the area amid a still-heavy presence of police.
French authorities had warned "radical activists" might disrupt the Paris demonstration Wednesday as in previous yellow vest protests and on May Day during the last two years.
Associated Press reporters saw groups of the masked and hooded protesters causing damage and then merging with the much larger number of peaceful May Day marchers.
Some vandalized a parked van, kicking the vehicle and breaking its windows. Others set small fires to trash cans and a shed at a construction site.
At least two men with head wounds were helped away by paramedics and firefighters assisted a woman in a wheelchair. Some police officers also fell on the ground.
Paris police said one police officer was taken to a hospital with a head injury.
The Russian Foreign Ministry alleged that French police used batons on the head and shoulder of a correspondent for state news agency RIA-Novosti, Viktoria Ivanova.
"We consider the use of violence against journalists in the exercise of their professional duties to be unacceptable," the ministry said in a statement on Wednesday.
The Paris police department and the French government did not immediately respond to the statement from Moscow.
While some of the people clashing with police wore the signature yellow vests of a French anti-government movement, the peaceful march also had participants in yellow vests as well as waving labor union flags.
Yellow vest supporters joined the traditional May Day march organized by labor unions to show solidarity in rejecting President Emmanuel Macron's economic policies, which some see as favoring the wealthy and big business.
Macron last week tried to address the complaints of the yellow vest movement by announcing tax cuts for middle-class workers, a pension increase and election rules to make it easier to call public referendums.
Philippe Martinez, secretary general of the CGT union, temporarily left the march for security reasons during the scuffles between the anarchist protesters and police.
After rejoining the marching throngs, he stressed that yellow vest and union activists "are marching together in all French cities."
"That's a protest of workers who tell the government and the president of the republic: 'Change your policies,'" Martinez said of the support from the movement that started in November. "We are very satisfied of the mobilization."
Signs held aloft during the march read "Long live freedom, long live socialism," ''Police, gendarmes, join us," and "What are we going to leave our children? Wake up."
French lawmaker Eric Coquerel, a member of far-left party France Insoumise ("Rebel France"), said "violence is, unfortunately, often playing against protesters." Larger numbers of demonstrators would be "more efficient" to put pressure on the government, he said.
French police ordered the closure of more than 580 shops, restaurants and cafes on the Paris protest route and numerous subway stations were shut.