In the final minute of four added on at the end of the match, home skipper Steven Schumacher was tripped in the box by Alex defender Mark Ellis. Bishop stepped up and buried the spot-kick past goalkeeper Steve Phillips.
Bury created the bulk of the first-half chances but ended up going in at the interval a goal behind.
That strike came in the 28th minute. A sweet Crewe move culminated with Kelvin Mellor teeing up teenager Max Clayton and he hooked home an angled shot to register his seventh goal of the season.
Earlier, Nicky Ajose shot straight at Alex keeper Phillips while Efe Sodje blazed over from close range.
Phillips also produced a brilliant save to keep out Ajose's volley from just six yards out.
Bury eventually levelled it up in the 59th minute. Ajose played the ball across the six-yard box to Tom Soares, whose shot on the turn squeezed past the unlucky Phillips.
Substitute Brad Inman fired Crewe back in front in the 83rd minute as he lashed home from 18 yards but Bishop kept his cool to ensure the points were shared.