Relations between the East African countries have often been tempestuous and Rwanda has previously denied claims it backs the rebels.
The interior ministry said RED-Tabara insurgents— accused of a string of attacks in Burundi— threw a grenade at a packed bus stop in the main city of Bujumbura at 7:00 pm (1700 GMT) on Friday.
"Thirty-eight people were injured, five of them seriously," the ministry's spokesman Pierre Nkurikiye told reporters.
"There were no deaths," he said, but police sources and witnesses told AFP at least three people had been killed.
In a simultaneous attack near a petrol station in the northern Ngagara district, only the attacker was wounded, Nkurikiye said, adding that he was swiftly arrested.
Nkurikiye, who was speaking at the National Intelligence Service headquarters in Bujumbura, paraded six suspects accused of being behind four attacks since April.
"These terrorists were recruited, trained and even equipped with weapons in Rwanda and by Rwanda," he said, citing their investigations.
"Afterwards, they are sent to the RED-Tabara terrorist group to be close to Burundi's borders, and then they enter the country to carry out attacks," Nkurikiye said.
The RED-Tabara emerged in 2011 and has a base in the eastern Democratic Republic of Congo province of South Kivu.
Burundi has repeatedly accused Rwanda of backing the rebels — a claim denied by the Rwandan government and RED-Tabara.
RED-Tabara is accused of waging deadly violence in Burundi since 2015 but had not been active there since September 2021, when it carried out several attacks, including on the airport in Bujumbura.
It is now the most active of Burundi's rebel forces with an estimated 500 to 800 fighters.
Burundi closed its border with Rwanda in 2015 amid mutual claims of supporting rebel groups.
The border was reopened in 2022.
In January 2024, Burundi said it had closed the border again after a rebel attack that killed 20 people, including women and children.