Both Yeovil keeper Scott Loach and Saddlers stopper Richard O'Donnell produced heroics by saving penalty kicks.
But unlucky Loach blundered in the game's key moment when he allowed Hiwula's 32nd minute shot to squirm under his body. In the end it was enough to earn the Saddlers all three points in a game created enough chances to win several times over.
Yet they relied on the brilliance of O'Donnell, who after keeping out Adam Morgan's second half spot-kick produced a remarkable save to keep out Byron Webster's late header.
The result moves the Saddlers back to within five points of the top six but more importantly ends the sudden slump which threatened to have them looking over their shoulders.
Smith made three changes to the team which was beaten so comfortably by Leyton Orient on Saturday, with only one enforced as injury saw Jordan Cook replaced by James Baxendale.
The manager also opted to rest Michael Cain in midfield, with Reece Flanagan handed a first start since October's 1-0 defeat to Crewe. Paul Downing was meanwhile returned to the centre of defence in place of James O'Connor, who dropped to the bench along with Cain.
Much of the first half narrative concerned the fluctuating fortunes of Loach, signed only 24 hours previously by the Glovers, who are suffering something of a goalkeeping crisis with both Chris Weale and Artur Krysiak out injured.
Yet the hosts were inches away from grabbing a lead inside three minutes as Gozie Ugwu's cross flashed across the face of the Walsall goal, just inches in front of striker partner James Hayter.
The home crowd were then incensed after no card was shown to Bradshaw for his part in an aerial challenge which left Stephen Kingsley needing lengthy treatment and forced the hosts into an early change, with Nathan Ralph coming on in defence.
Hiwula wasted a great change to increase their anger when he pounced on a loose ball and raced into the box but with the defence backing off, he could only shoot wide.
Yet the youngster's pace would soon present the Saddlers with another golden chance as he beat Loach to a loose ball and was promptly brought down by the keeper.
Bradshaw had failed to score in his previous five appearances - his longest drought since joining the club. And he squandered the chance here, sending a low spot-kick almost straight into the arms of Loach.
Walsall goal: Hiwula 32'
Walsall: O'Donnell, Purkiss, Downing, Chambers,Taylor, Chambers, Flanagan (Cain 89'), Baxendale (Grimes 78'), Sawyers, Hiwula, Bradshaw.
Subs not used: O'Connor, MacGillivray, Preston, Bakayoko, Kinsella
Yeovil: Loache, Moore, Kingsley (Ralph 12'), Webster, Shephard, Foley, Berrett, Sheehan, Davis, Hayter (Morgan 58') Ugwu
Subs not used: Hoskins, Ofori-Twumasi, Grant, Stewart, Eastmond
Ref: Charles Breakspear
Yet the keeper's joy at swift redemption was snatched away just four minutes later when he allowed Hiwula's low drive to squirm under his body and into the net, after the forward had again been given the chance to charge at the home defence.
Still, Loach went some way to making up for that error soon after by keeping out a rasping Romaine Sawyers drive as the game entered a period of almost total Walsall domination.
James Baxendale too should have got better contact on a far post shot which the keeper was able to push round the post. There was no doubting the sweet contact Hiwula got on a shot soon after but again Loach proved equal to it, flinging himself to the left to keep it out with his fingertips.
The only wonder was how the Saddlers weren't winning by more and they almost saw their advantage wiped out in first half stoppage time, when a calamitous free-kick routine led to a Yeovil break in which only a key Adam Chambers tackle on Liam Shephard likely prevented a leveller.
The breathing space the Saddlers craved almost arrived early in the second period as Baxendale was allowed to race within 12 yards of goal before unleashing a shot which cannoned off a defender and then the post before being hacked clear.
They then received another warning of the thin nature of their advantage when Ugwu had the ball in the net, only to be denied by a linesman's flag.
Yet still the visitors appeared reluctant to take advantage of the numerous chances coming their way, with Hiwula and Sawyers shooting off target from good positions.
Hiwula then failed to take advantage of a superb lofted Flanagan pass which he could only lift well over the bar, while a succession of corners provided the hosts first real sustained pressure of the night.
It led to a chance for Ugwu, who was superbly denied at close range by O'Donnell after a deep cross found him unmarked at the far post.
The keeper was soon required for further heroics as he kept out Morgan's spot-kick diving to his left, after Ben Purkiss had been penalised for bringing down Joel Grant.
Substitute Ashley Grimes thought he had eased any Walsall nerves with a shot which looked destined for the bottom corner before Loach appeared to push it wide.
O'Donnell then kept out Josh Sheehan's drive before somehow clawing Webster's header from the corner away.
The keeper then saved from Morgan deep into stoppage time to avoid what would have been a maddening draw.

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Walsall vs Yeovil action. Pictures courtesy of Kevin Grice

