The on-song hitman netted twice in the space of five minutes either side of half-time as Alan Knill's relegation-threatened side snatched three welcome points at Gigg Lane.
Stewart Drummond pulled a goal back for Morecambe two minutes after the visitors had gone 2-0 down, but Bishop's goals - his fourth in two games - proved the difference between the teams.
It was Bury who were the brighter starters and the home side created a string of first-half chances.
But the breakthrough deservedly came just as fourth official Colin Harwood was about to hold up his board for time added on.
Nicky Adams found Bishop with a delightful left-wing cross and Bishop just managed to poke the ball beyond Shwan Jalal sliding in at the back post.
Then in the 50th minute Bury doubled their advantage.
Carl Baker lost the ball in his own half and, tracking back with tired legs, felled Adams in his own penalty for a clumsy penalty.
Bishop stepped up and made no mistake from the spot, while Baker was immediately substituted and he stormed off down the tunnel.
However, the home side's joy was shortlived as Drummond headed past Darren Randolph from Garry Thompson's excellent corner.
Bury dug in and defended well, but David Artell had a good chance to level when Morecambe won a 77th minute free-kick on the edge of the area.
However, the defender fired his curling right-foot shot high and wide much to the disappointment of the 367 away supporters.