A grandfather was seriously injured during the "terrifying" incident at St Patrick's Cemetery in Co Louth, the parish priest said.
A man in his late 20s, who police believe was the driver of a dark car involved in the collision, was arrested at the scene in Dowdallshill, Dundalk, on Sunday afternoon.
The injured elderly man was taken to Our Lady of Lourdes Hospital in Drogheda, while a "small number of people received superficial injuries", gardai said.
Fr Mark O'Hagan said thousands had gathered for the annual blessing of the graves service in the cemetery when shouts and screams erupted as people frantically jumped out of the way of the speeding car.
Footage posted online showed emergency services at the scene as crowds looked on, while pictures later emerged showing a topless man being taken away by officers.
Fr O'Hagan told PA: "I could hear this screaming and shouting. And there was a lot of people running and a lot of people screaming."
He described the car speeding through the graveyard past him towards a car park before driving back in his direction and mounting the kerb.
He added: "I tried to wave at him to try and slow him down or stop him.
"He sped on straight through the people again. It was terrifying."
Sinn Fein councillor Ruairi O Murchu, who was not at the event but spoke to some people who were, said those present were "utterly shocked" and "angry" about what had happened.
He said: "This ceremony is attended by thousands of people every year, including many children and elderly people, who come to Dowdallshill to remember loved ones.
"People who were very close to where the car sped through were jumping aside."
Police said the car collided with parked vehicles as it was driven out of the cemetery, and the suspect was arrested once it had come to a stop on the road.
Gardai said: "A man in his late 20s, believed to the driver, was arrested at the scene. He is currently detained at Dundalk Garda Station under the provisions of Section 4 - Criminal Justice Act 1984."
The force appealed for witnesses, especially anyone with video footage of the incident, to contact them. Dundalk Garda Station can be contacted on 042 9388400, the Garda Confidential line 1800-666-111 or at any other station.
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