After three consecutive league defeats on the road had put considerable pressure on Owen Coyle, the Whites could ill-afford another off day away from the Reebok.
And what they produced was a solid, and often inventive performance, which ranks by far their best of the season so far.
Wanderers shaded the first half after a slow start, and went in a goal to the good thanks to a fine header from Alonso, who got in front of marker Nejc Pecnik to head Chris Eagles' corner past Chris Kirkland.
Coyle's side had improved as time went on, but were perhaps lucky not to be behind after Rhys McCabe's low free-kick had sneaked under the wall and forced Adam Bogdan into a good low save, albeit one with the help of his post.
A couple of poor choices by Jay Bothroyd then helped Wanderers keep on level terms but they grew into the game and manufactured a few chances of their own.
Eagles missed two good chances from close range and David Ngog also headed over the top from a bouncing Keith Andrews effort.
But Alonso's header had given the Whites a slender advantage and it was perhaps a concession to the number of goals they had conceded after the interval this season that the players returned early to the pitch to complete a warm-up.
It appeared to have worked, and after controlling the tempo, Ngog should have doubled the lead when Alonso rolled a perfect cross into the six yard box only for the Frenchman to fail to get a telling touch.
But it all came tumbling down on the hour mark when Keith Andrews went to ground in the box, and Michail Antonio wasted no time dropping to the floor.
Referee Keith Stroud pointed to the spot, and on-loan Everton midfielder Ross Barkley calmly slotted the penalty into the bottom left hand corner to level the scores.
Now would be the challenge for Wanderers, who could easily have collapsed at that point. But fans need not have worried - it took just three minutes for them to regain their lead.
Ngog hassled a mistake out of Anthony Gardner and when Kevin Davies played in his namesake Mark, the midfielder scampered around two would-be challengers before crashing a fine shot into the top corner.
Wanderers survived another penalty shout moments later when a Martin Taylor header seemed to catch Tyrone Mears on the hand – and had another lucky escape when Pecnik drilled the resulting rebound inches wide of the post.
But for the home side to pull themselves level again wouldn't have truly reflected the balance of play, and the Whites managed to hold firm until the end to register a vital win.