Bunting looked set for a straightforward night when he ripped through the opening phase with a 119 average for the first set, backing it up with a clinical 121 checkout and then a brutal statement finish to take control.
He then produced arguably the shot of the evening, taking out 160 on tops, described on the broadcast as a "monster of a finish", as he pushed towards a commanding lead.
Bialecki, though, refused to accept the script. As Bunting's level dropped from its opening peak, the Pole found enough scoring and composure to punish, taking the fourth set 3-0 and winning three legs on the spin after breaking early.
For a stretch, world number four Bunting went from unplayable to merely mortal, which at Alexandra Palace is often the moment the match starts playing the player rather than the other way round.
The key for Bunting was stopping the slide before it became terminal.
After being pulled into a scrap, he steadied, regained control of the tempo and worked his way through the danger period to get over the line and book his place in the next round via a 3-2 victory.
'Got away with it'
"Playing at Ally Pally you go through all the emotions," Bunting told Sky Sports.
"I was lucky to get away with that one. I need to regroup, get back on the practice board, and get ready for the next game.
"I heard Toby shouting my name, I have got great support behind me, great family.
"Even the Ally Pally wasp came back from the brink.
"These crowds are amazing. Let's go Bunting Mental!
"That is the nervous one out of the way. I promise I will be better in the next round."
Bunting will face Nitin Kumar in the next stage, with the Indian making history in the afternoon session after becoming the first player from the country to win a match at the World Championship.
Ven den Bergh stunned
Earlier in the session, Wesley Plaisier beat Lukas Wenig 3-1 in sets, producing a decisive surge that included five perfect darts at the start of a leg as he teased a nine-darter, before closing out the match on double 9.
The Dutchman could face Gerwyn Price in round two, should The Iceman come through his opener against Adam Gawlas on Tuesday.
The biggest upset came as debutant Darren Beveridge sent world number 23 and former semi-finalist Dimitri Van den Bergh out in straight sets, exploiting a night of missed doubles from the Belgian, who was playing televised darts for the first time since April after taking time away to focus on his health and wellbeing.
James Hurrell then closed proceedings by beating Stowe Buntz 3-1 in a tense finale, returning to double 16 to settle the deciding set after Buntz missed opportunities to land a late break and could not take out 152 when it mattered.
The action continues on Monday in north London, with two-time champion Peter Wright, Cameron Menzies, Jonny Clayton and a returning Simon Whitlock all in action.