But Daniels parried Jevons' weak spot-kick and reacted quickly to block Mike Jones' attempt to force home the rebound.
That seemed to galvanise Shrewsbury, who went close when McIntyre dragged a shot wide and Paul Murray's 30-yard drive was tipped over by Bury keeper Wayne Brown.
Brown also produced a fine save just before half-time, pushing over Ben Davies' free-kick.
Shrewsbury failed to maintain that momentum after the break and had a lucky escape when Elliott Bennett's low cross just evaded Morrell, who seconds later had a shot saved by Daniels.
But just as Ashton's first-leg mistake looked like being decisive, he sent a long ball into the Bury box. And, after David Buchanan failed to clear the danger, McIntyre netted from 15 yards to dramatically rescue his side - the midfield man's first goal since October in only his third game back after the broken leg which had looked likely to end his season in January.
After that lift, second-half sub Leslie was then controversially red-carded less than a minute after the start of extra time and Shrewsbury found themselves on the back foot once more.
Neil Ashton (right) gave away a penalty by fouling Andy Morrell
But Daniels again came to their rescue, superbly denying Bishop after the forward had been put through on goal, and by diving to his left to push away Buchanan's rasping shot.
Bury kept up the pressure and Shrews defender Darren Moss cleared Efe Sodje's header off the line, before Bishop inexplicably failed to connect with Bennett's right-wing cross.
Bishop was then denied by two more brilliant stops from Daniels, and Bury appealed in vain for a penalty after Bennett went down under a challenge from Ashton.
In the dying seconds, Bury's Ben Futcher shot wide from close range and Bennett was thwarted by yet another save from the outstanding Daniels.
It came as no surprise when Daniels' heroics continued in the shoot-out.
Shrewsbury converted all of their penalties, with Davies, Moses Ashikodi, McIntyre and David Worrall hitting the target, while Jones and Brian Barry-Murphy scored for Bury.
But Racchi was unable to find a way past Daniels, who then saved Bishop's kick to seal a second trip in two years for Shrewsbury to Wembley, where the beaten 2007 play-off finalists will face Gillingham.
Bury manager Alan Knill: "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."
Shrewsbury manager Paul Simpson:
"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."
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