Allen produced his brilliant frame in a second-round match against qualifier Wakelin at Sheffield's Crucible Theatre.
The 39-year-old punched the air after sinking the final black in the 13th frame and was warmly congratulated by Wakelin.
Allen is only the 11th player to make a 147 at the famed Crucible, while it was the 15th maximum in the main draw of the tournament and the first since Mark Selby in the 2023 final.
Allen will receive a £45,000 prize for recording the 217th maximum in snooker history.
Seven-time world champion Stephen Hendry told the BBC that Allen's clearance was "absolutely phenomenal" and added: "What a performance. Unbelievable. Incredible from Mark Allen.
"He was 10-2 behind and he's not scored a point for three frames, comes out after the interval and he makes a 147."
But Allen's joy was short-lived. Wakelin raced into a 12-4 lead, and despite Allen winning the next two frames, eventually took the match 13-6 to book his place in the quarter-finals.
Wakelin will next face the winner between Lei Peifan and Xintong Zhao, with Zhao leading 5-3.
In the evening's other session, Mark Williams leads 9-7 over Iran's Hossein Vafaei, with the winner set to face either John Higgins or Guodong Xiao - which is poised 8-8 - in the last eight.
All three sessions will resume on Saturday.