Bury were reduced to 10 men after the first of Andy Monkhouse's two goals when Fagan was guilty of serious foul play on Pools' left-back Darren Holden.
Monkhouse then snaffled a second goal in the 72nd minute to lift Pools off the bottom by leapfrogging the Shakers.
The visitors rarely troubled Pools, going close in the 55th minute when striker Andy Bishop fired over on the turn.
Manager Kevin Blackwell said: "We should have been comfortable 2-0 up and conceded a soft goal. We've never capitalised on our play, it would have been a walk in the park and we made it hard for us.
"Nicky Ajose should have scored when he was put through when it was 0-0, you dream of chances like that. Not half-chances, clear chances.
"It's the old adage that goals win games, we go one-up, and they were all over the place."
On Fagan's red card Blackwell added: "Fagan's sending-off was rash - no excuses. Some strange decisions, but that was a red card.
"He had a long chat with the linesman and then decided. We kept going, had pressure at the end, but lack quality.
"It's been such a long season, the whole situation is grim."