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After Bradshaw found the bottom right corner to cancel out defender Mark Ellis's early first-half opener, James Collins put Shrewsbury back in front.
His shot from outside the box looked to have avenged Shrewsbury's 3-0 Cup first-round exit at Walsall a year ago.
But Bradshaw levelled again to rescue the Saddlers in the 90th minute.
He deflected in a shot from Paul Downing for his ninth goal of the season just when the League Two side, who put out three higher-ranked teams on their way to this season's League Cup fourth round, were scenting another upset.
Town led inside five minutes when an inswinging corner from Liam Lawrence was only partially cleared and big defender Ellis slammed a low drive into the net from 10 yards.
Bradshaw then equalised in the 56th minute with a low drive from Romaine Sawyers' pass, only for Shrewsbury to regain the lead five minutes later, when Collins deflected in Andy Mangan's free-kick.
But, after Town keeper Jayson Leutwiler had gone off injured, Bradshaw popped up again to book his return to the Meadow on Tuesday week (18 November, 19:45 GMT).
Walsall manager Dean Smith told BBC WM:
"It was nice to see Tom Bradshaw bag a brace. He is a level-headed lad who works really hard. He was shackled well in the first half but good players persevere.
"We were the better team over the 90 minutes. We conceded an early goal, which was disappointing, but we took the game to them after that.
"We could have done a bit better in the final third, but we deservedly got ourselves back into the game. It would have been a travesty had we not been in the second round draw."
Shrewsbury manager Micky Mellon told BBC Radio Shropshire:
"I am very disappointed to have conceded a goal at the very end. I can't remember Walsall having a single chance in the build up to it.
"But we are in the hat for the draw on Monday. And we'll have a chance when we take them back to the Meadow - we'll know what the prize is.
"We've had a few changes forced upon us but, if you'd have said we'd score twice here at Walsall, I'd have said that would have given us a good chance."
Walsall top scorer Tom Bradshaw scored twice against former club Shrewsbury Town to earn Dean Smith's Saddlers an FA Cup first round replay.