Separately, two gunmen riding on motorcycles opened fire on troops at a military checkpoint in Kano on Sunday, injuring a soldier, military spokesman Iweha Ikedichi said.
After the blast late Sunday at the school in the Yobe state town of Damagun, which destroyed part of the building, officers deployed around the city, police spokesman Toyin Gbadegesin told AFP.
Police then intercepted the militants as they tried to storm the church and the station early Monday, he added.
It was not immediately clear who was responsible for the attacks, but they resembled those carried out by Boko Haram.
“At about 1:05 am (0005 GMT) attacks launched against Damagun police division and Catholic church were successfully repelled by the police, who were on full alert,” Gbadegesin said.
He reported no casualties from the three incidents.
Damagun is roughly 30 kilometres (20 miles) from the Yobe state capital Damaturu, a city hit hard by Boko Haram which last week saw gun battles between officers and suspected Islamists that left at least two people dead.
Security has been tightened around the country for the Eid festivals, which mark the end of the Muslim fasting month of Ramadan.
The Eid al-Fitr celebrations began at the weekend, but Monday and Tuesday have also been declared public holidays.