Bury laid siege to the Shrews' goal after the visitors had sub Steve Leslie sent off in extra-time but were denied by a string of outstanding saves by the on-loan West Brom goalkeeper.
The 21-year-old then saved from Bury's Danny Racchi and top scorer Andy Bishop in the penalty shoot-out, in which the Shrews prevailed 4-3 to book their ticket to Wembley.
Phil Jevons had earlier missed a 31st-minute penalty, which would have given Bury a 2-0 aggregate lead.
The Shakers, who finished just one goal short of automatic promotion, still looked set to go through before Kevin McIntyre broke Bury hearts with an 88th-minute strike to send the game into extra-time.
Andy Morrell was preferred to Bishop, who has been struggling with a knee injury, and Morrell had the first effort on target from a Brian Barry-Murphy pass.
Shrews left-back Neil Ashton, scorer of the late own goal that decided the first leg, then crossed for Grant Holt to volley over from 16 yards.
Morrell tested Daniels again after linking well with strike partner Jevons, before Kelvin Langmead almost bundled in a McIntyre corner.
Bury's Stephen Dawson then sliced a 20-yard effort wide while visiting striker Omer Riza almost latched onto a risky backpass by Efe Sodje.
Then came Bury's first penalty blow as Morrell was brought down by Ashton and Jevons' weak spot-kick was saved low to Daniels' right.
Daniels then blocked the rebound from Mike Jones with his feet and saved Dawson's header to send the sold-out away end into raptures.
Wayne Brown gave the Shakers a scare as he misjudged a long ball and Holt turned the ball wide from a tight angle, before Jevons was unable to keep the ball down after Ben Futcher nodded down Jones' corner.
The Shrews finished the half strongly with McIntyre firing wide after being put through by Holt while Brown had to tip over a 25-yard strike by Paul Murray.
Brown then pulled off a superb save in first-half stoppage time to turn over a free-kick by Ben Davies.
Murray had a shot blocked shortly after the restart before Daniels made another fine save to deny Morrell from close range.
Jevons was then unable to direct Elliott Bennett's right-wing cross goalwards and Bury had another penalty shout turned down as Jones went down under Dave Worrall's challenge.
Bury sent on Bishop and fellow striker Glynn Hurst but the Shrews remained a threat, almost bundling in a Davies corner while Holt tested Brown with a 20-yard volley.
The hosts were pushing forward more late on, though, and looked set to seal it until Dave Buchanan failed to clear and McIntyre pounced to smash home a left-footed volley from 14 yards.
Bury 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.
The Shrews keeper then parried a Buchanan strike before Sodje headed a Jones free-kick goalwards and Darren Moss headed the ball behind from under his own bar.
Daniels denied Bishop three times in the second period of extra-time and also saved with his feet from Bennett, before pulling off his shoot-out heroics.