The attacker targeted St Polycarp’s church on Juja Road.
A police spokesman blamed sympathizers of Somalia’s al-Shabab Islamist militant group, angry over Kenya’s role in the UN-backed intervention force.
A mob later rounded on Somalis living near the church with sticks and stones in a suspected revenge attack.
Police chief Moses Nyakwama said 13 people had been injured in the revenge attack, in the suburb of Eastleigh.
Reports suggested a number of those hurt at the church were injured in a stampede after the attack.
A police spokesman, Charles Owino, told Reuters news agency: “We suspect this blast might have been carried out by sympathizers of al-Shabab.
“These are the kicks of a dying horse since, of late, Kenyan police have arrested several suspects in connection with grenades.”
The authorities said three children were seriously hurt in the attack, and a number of others suffered lighter injuries.
The Red Cross had earlier said six children were critically wounded.
Irene Wambui, who was in the church at the time of the attack, said: “We were just worshipping God in church when suddenly we heard an explosion and people started running for their lives.
“We came to realize that the explosion had injured some kids who were taken to hospital and unfortunately one succumbed.”
Senior Nairobi police officer Moses Ombati appealed for calm after youths reportedly attacked the nearby Alamin mosque.
Nairobi and the port city of Mombasa have suffered a series of grenade attacks since Kenya sent troops into Somalia last October.
The attacks in Mombasa escalated after radical Islamist preacher Aboud Rogo Mohammed was killed in a drive-by shooting in August.
In July, 15 people were killed in raids on churches in Garissa, near Kenya’s border with Somalia.
There was speculation that al-Shabab or its sympathizers were responsible.
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