The feat places Murphy among a select group of snooker greats, becoming the fifth player to achieve snooker perfection – clearing all 15 reds with 15 blacks, followed by all the colours – since the tournament's inception in 1975.
The 42-year-old, Masters champion in 2015, trailed 2-0, with Allen dominating the early tactical exchanges.
Murphy soon shifted momentum in the third frame and eventually levelled the score. He then embarked on a four-frame winning streak, culminating in the historic 147 break in the sixth frame, extending his lead to 4-2.
The maximum break, executed with impeccable cue-ball control, ignited jubilant scenes within Alexandra Palace, including from his opponent.
Murphy described it as "an incredible moment" after the match.
"I can't believe it. It was an incredible moment, one of the best of my snooker career," Murphy told the BBC.
"I have wanted to make a 147 in one of the BBC events since I was a child. I had a great opportunity the other day and totally messed it up.
"I had great support from the crowd, who were absolutely amazing."
While Allen briefly reduced the arrears with a timely half-century break, Murphy applied ample pressure with a 72 break in the eighth frame before clinching the match in the ninth after a crucial safety exchange on the green.
Murphy, known as "the Magician," now boasts nine 147 breaks throughout his career, trailing only Ronnie O'Sullivan (15), John Higgins (13), and Stephen Hendry (11) in this elite category.
His 147 in the Masters puts him in contention for the £15,000 prize awarded for the tournament's highest break.
He will face either the winner of Saturday evening's second semi-final between Judd Trump and Kyren Wilson in Sunday's final.