The Shakers, who finished just one goal short of automatic promotion, were two minutes away from Wembley as they held a one goal advantage from the first leg but Kevin McIntyre's 88th-minute strike broke Bury hearts and sent the game into extra-time.
Bury then laid siege on the Shrewsbury goal after they had sub Steve Leslie sent off but were unable to make the breakthrough.
The Shrews prevailed 4-3 in the penalty shoot-out with Danny Racchi and top scorer Andy Bishop missing the crucial spot-kicks. Phil Jevons had earlier missed a 31st-minute penalty, which would have given Bury a 2-0 aggregate lead and put them on the verge of their first trip to Wembley since 1995.
Omer Riza almost latched onto a risky backpass by Efe Sodje, (where have I written that before), then came Bury's first penalty blow after Morrell was brought down by Ashton. Daniels saved Jevons' weak spot-kick low to his right before denying Mike Jones from the rebound and saving Dawson's header.
Bury re-emerged for the second half with greater urgency and Daniels had to make another fine save to keep out Morrell's close-range strike.
Jevons and Jones both went close before Knill made a double substitution, Bishop and fellow striker Glynn Hurst coming on in a bid to seal Bury's ticket to Wembley.
Holt tested Brown with a 20-yard volley but Bury looked comfortable until Dave Buchanan failed to clear and McIntyre pounced to smash home a left-footed volley and stun the Shakers.
Knill's men were handed a massive boost just seconds into extra-time as Leslie was sent off for elbowing Dawson, and moments later, Steve Haslam's long ball put Bishop clean through but his finish was too close to Daniels.
He then denied Buchanan and Darren Moss turned a Sodje header over his own bar before Bishop glanced a superb Bennett cross just wide.
The inspirational Daniels, on loan from West Brom, denied Bishop three times in the second period of extra-time and also saved with his feet from Bennett.
Daniels then saved from Racchi and Bishop in the shoot-out to draw the curtain on what has still been a great season for Bury but they'll be no Sodje again at Roots Hall next time around.
Will we ever see him in action against the Blues again as he turns 37 next October?
Bury manager Alan Knill said after the game: "We kept opening them up and I thought sooner or later we were going to score. We have had clear-cut chance and if you don't put them away, you are not going to win games. We did everything we had to do, but couldn't put the ball in the back of the net."
Meanwhile Shrewsbury manager Paul Simpson replied: "Maybe our goal lived a charmed life today but that was because Luke Daniels worked so hard and produced some great saves. We got it from sheer guts and determination and showed unbelievable composure at the end. We got level on 88 minutes and I hoped we could just nick it, but penalties are all right if you win."
Bury: Brown, Haslam (Racchi 112), Futcher, Sodje, Buchanan, Bennett, Barry-Murphy, Dawson, Jones, Morrell (Bishop 70), Jevons (Hurst 70).
Subs not used: Cresswell, Welsh.
Booked: Bennett, Futcher, Dawson 85, Sodje 90.
Shrewsbury: Daniels, Darren Moss, Coughlan, Langmead, Ashton, Worrall, Murray (Leslie 66), Davies, McIntyre, Riza (Ashikodi 71), Holt (Chadwick 107). Subs not used: Dunfield, Herd.
Goal: McIntyre 88.
Booked: Davies
Sent off: Leslie
Referee: Colin Webster (Tyne & Wear)
League Two Play-Off Final (at Wembley):
Saturday 23 May 2009, 15.00.
Shrewsbury v Gillingham