The Dutchman ignited at the end of an undulating tie in London, rattling off three ton-plus finishes in the final six legs of the clash, which went all the way to a deciding set.
Doets, who will next face former world champion Luke Humphries, capitalised on some wild finishing from Aspinall to secure a sixth straight win against his opponent.
"That was insane!" Doets told Sky Sports. "I started off rough. I was lucky to be 2-2, but then I should have been 3-2 up. He hit the 170 to go 3-2 up, but after that I was just on fire, I couldn't miss any more.
"This is what you dream of. This is my second year in a row in the last 16 and I want to go even further."
Aspinall settled down instantly with a 102 checkout and a couple of holds later, he dealt the first blow in the match after pressuring Doets' throw.
A tight second set was taken to a deciding leg and both players squandered chances to seal, but it was Doets who managed to hold his nerve on double eight to draw level.
Aspinall had a chance to throw out for the third but misfired, allowing Doets to come back and remove 106 to force a decider. 'The Asp' would defy that wobble, though, claiming the set to move back ahead.
He built a 2-0 lead in set four but relinquished it again, with Doets taking out 121 to send it to another decider. It was the Dutchman's turn to sneak it this time, making it 2-2 with Aspinall missing his opportunities.
Aspinall steadied himself by checking out a match-high 150 in the fifth and a fourth straight deciding leg was required, where the former prevailed by reeling in a clutch Big Fish 170 finish.
Yet Doets came back firing again, rattling in back-to-back 112 and 164 checkouts to send a pulsating match all the way to a final-set shootout.
The scoring barrage from Doets continued as he opened the deciding set with a 100 out, followed up with back-to-back double 19s to check out then hit double 10 to complete the scalp.
Manby continues dream debut
Charlie Manby battled some finishing trouble to beat a wilting Ricky Evans 4-2 and set up a fourth-round clash with 10th seed Gian van Veen.
A cagey opening set which saw both players average in the 80s fell the way of Evans, who boasted a checkout rate just shy of 43 per cent.
The pair raised the tempo in set two and despite 20 missed doubles to this point, Manby equalised to make it 1-1 in a decider.
'Rapid Ricky' pounced on some more wayward finishing from Manby in stunning fashion by removing 134 to start the third, which sent him on his way to open a 2-1 lead.
Debutant Manby, though, came back to edge a fourth-set decider, squaring things at 2-2 even with 18 per cent on the doubles.
Evans' scoring plummeted to a 73.80 average in the fifth set and that allowed Manby to seize the initiative, reeling off six straight legs to complete the comeback.
Hood rolls into last 16
Debutant Justin Hood continued his impressive run with a 4-1 win over Ryan Meikle in a contest which split six ton-plus checkouts between them.
Hood saw his throw broken in the first leg of the match with a 100 from Meikle, but he replied with a 102 of his own to grab the decider.
The pair exchanged breaks in the second and 'Happy Feet' doubled his lead, managing to avoid being stuck in the madhouse with another deciding-leg win.
Hood whitewashed set three to make it 3-0 and took out 164 in the fourth, but Meikle got his first on the board on the back of a stunning 147 checkout.
That proved to be a consolation, however, as Hood ran out the winner of another deciding leg to complete a solid victory.
"It's good, but there's still more games to go," Hood told Sky Sports.
"I'm playing well. Today wasn't my best performance, but I'll put that right next game."
Monday's 2026 PDC World Darts Championship results
Afternoon session
Nathan Aspinall 3-4 Kevin Doets