The April 29 attack on the car carrying the cardinal occurred around 5 p.m., as he was en route to Uromi.
Onaiyekan told Vatican Radio May 2, "We were in the wrong place at the wrong time."
He said he had tried to avoid speaking publicly about the incident "because they were not targeting me. The car I was riding in was on a public road together with many other cars. We ended up in the middle of one of the attacks that take place every now and then. As the saying goes, we were in the wrong place at the wrong time."
Such attacks have become frequent, the cardinal said.
"Two or three weeks ago a car in which two or three priests were riding was attacked and one of the priests was seriously injured."
The cardinal said he could not identify the gunmen and did not know for certain they were Fulani, nomadic Muslim cattle herders.
"We didn't see anyone, we just heard the 'bam, bam, bam' of the guns and saw that a bullet pierced the car. I didn't see anyone. It could have been them (the Fulani), it could have been other criminals, but in the end what matters is that the road is not safe. We cannot go around with an armed escort. That is no way to live.
"I do not believe they were targeting me or that they knew I was in the middle of that chaos," the cardinal said. "In fact, I imagine that what happened was embarrassing for them because up until now they have attacked people without drawing much attention.
"It was a criminal act," he insisted, not one motivated by religion because "in all those cars on the road that day there was no way to know if someone was Muslim or Christian."
The day after the attack, Akubueze told newsmen in Benin City, "If something bad had happened to him, what would have been the story? ... We thank God nothing happened and we are using this opportunity to appeal to Mr. President to provide adequate security for the citizens of Edo state."
"If there is no security of lives and property, no investments or meaningful economic development can take place in the state and the nation at large. We need security and we are demanding that from our government," he said.
The archbishop said the spate of attacks by the so-called herdsmen across the country "is becoming frightening."
Ogun said Nigerians, especially in Edo state, "cannot continue to live in fear in our own land because of the activities of these criminals."
Onaiyekan's driver, who identified himself only as Segun, said he slowed for a pothole.
"I would have passed the spot with speed, but because the cardinal was sleeping, I decided to slow down the vehicle so that he won't wake up," he said.
Three men emerged from the local palm tree plantation and started shooting at the vehicle," he said. Since only one was behind the car, he threw it into reverse and backed up.
When he finally stopped the vehicle, they discovered "that the bullets had shattered the left passenger's window glass and made huge holes on the panel of the door."