The promotion battle was one of only two League Two fixtures on Good Friday and Bury took full advantage to go two points clear of Wycombe and Exeter.
Phil Jevons gave Bury an undeserved half-time lead with a 45th-minute penalty and although Nick Chadwick levelled in the 55th minute, Hurst pounced ten minutes from time.
Town began the game in the final play-off place, six points adrift of Bury in fourth, and the visitors started the strongest in a lively start which saw the first chance in the 10th minute.
Shrews winger Chris Humphrey swung in a fine cross from the right, but top scorer Grant Holt headed the ball back across goal and wide.
His strike partner Chadwick then fired wide from inside the box, before Chadwick and Bury defender Efe Sodje required treatment for a clash of heads.
Bury goalkeeper Wayne Brown had to pull off a one-handed save low to his left to deny Holt after a fine run by Ben Davies, who then tested Brown with a 20-yard strike on the half-hour mark.
The best Bury could muster was a blocked shot from Brian Barry-Murphy, but they were awarded a penalty on the stroke of half-time.
Andy Morrell beat Graham Coughlan to the ball inside the box and on-loan striker Jevons stepped up to sidefoot the spot-kick home.
Shrewsbury got a deserved equaliser within ten minutes of the restart, though, as Chadwick rose above Steve Haslam to head home Davies' right-wing cross.
Holt and Bury skipper Stephen Dawson then fired well wide before substitute Andy Bishop, who had missed Bury's last two games through injury, tested Luke Daniels from distance with 20 minutes left.
But with time running out, Bishop flicked on Sodje's long ball and fellow sub Hurst nipped in front of Daniels to lift the ball over the Shrews' keeper and into the net.
Davies fired inches wide as the visitors pressed for a point, but Bury held on.