After exiting at the semi-final stage for the last three seasons, Pep Guardiola's treble-chasing side were not about to suffer the same fate and won without requiring top gear.
Championship promotion hopefuls United began brightly and frustrated City for most of the first half but Mahrez fired City ahead from the penalty spot in the 43rd minute.
Mahrez then waltzed through some flimsy defending to make it 2-0 in the 61st minute and completed his hat-trick five minutes later with a clinical finish from Jack Grealish's low cross.
It was the first FA Cup semi-final hat-trick since Alex Dawson for Manchester United against Fulham in 1958.
"The quality of Riyad we have seen for many years in this country, also with Leicester," Guardiola said of a player not always guaranteed a starting place at City.
"He is a huge mentality competitor. Huge ambition to score goals. Scoring at the end of the first half was so important."
City have not conceded a goal and scored 17 in reaching their 12th FA Cup final in which they could face Manchester United in what would be the showpiece's first Manchester clash.
United are up against Brighton & Hove Albion on Sunday.
With the tie safely in the bag, Guardiola substituted Grealish and goal machine Erling Haaland with one eye on Wednesday's looming Premier League showdown with Arsenal.
City are also in the semi-finals of the Champions League where they will face Spanish holders Real Madrid.
It proved a disappointing day for Sheffield United who last won the FA Cup almost 100 years ago, although the Yorkshire club's priority will be sealing promotion back to the top flight after being relegated in 2021.
City's Guardiola spoke of the fatigue of his side's relentless fixture schedule in the build-up to the semi-final.
But apart from a couple of anxious moments in the opening minutes, one of which saw Iliman Ndiaye denied from close range by the legs of City's second-choice goalkeeper Stefan Moreno, this was about as stress-free as it gets.
The gulf in class between the two sides was summed up by City's 79 per cent possession and by the fact that Moreno was largely a spectator against a side second in the Championship.
United thought they would arrive at halftime level with the English champions but a poorly-timed tackle by Daniel Jebbison on Bernardo Silva gave referee Stuart Attwell an easy decision.
Mahrez slotted the kick home to send City in ahead and the Algerian could hardly believe his luck just past the hour when he picked the ball up in the centre circle and sauntered through unchallenged to shoot past Wes Foderingham.
Grealish then dazzled United's defence before cutting back a pass for Mahrez to convert.
"It is an honour to be in the FA Cup final on Jun 3. We will be here. Today we played really good, everyone was serious," Guardiola, whose side can seize control of the title race if they beat Arsenal on Wednesday, said.
As a spectacle it was a disappointing affair and while Sheffield United's fans made the most of their day, the thousands of empty seats in the huge stadium gave more ammunition to those who disagree with semi-finals being played at Wembley.
