The 19-year-old fly-half, son of Saracens head coach and former Great Britain rugby league captain Andy Farrell, delivered an immaculate display.
Farrell junior landed five penalties and a conversion from six kicks at goal, adding the extras to wing James Short's 29th-minute try.
But he also made a telling tackle on his oppsite number Toby Flood when Leicester exhaustively went through more than 25 phases of play during the closing seconds.
Leicester saw their Aviva Premiership title bid flounder on two missed second-half penalties from Flood - kicks he would normally have expected to convert.
Flood still landed six penalties, yet his failures in quick succession midway through the third quarter undermined Leicester's attempts to remain in a game when they only briefly threatened territorially.
Saracens' last major trophy was the Tetley's Bitter Cup when they beat Wasps 48-18 in 1998 thanks to the efforts of world superstars like Michael Lynagh and Philippe Sella.
But an English league title was 135 years in the making from their formation, made all the sweeter against the revered kings of Premiership rugby.
Leicester, despite finishing top after the 22-game regular league season, never threatened to hit top gear during a match they began seeking a title hat-trick for the second time.
And Saracens now become England's fifth Premiership champions since the competition's induction in the 1997-98 season.
They follow seven-time winners Leicester, four-time champions Wasps, plus Sale and Newcastle, but Saracens now look set to march on and become, possibly, English rugby's next major force.
Leicester made one enforced change from the side that eclipsed play-off opponents Northampton two weeks ago, with suspended centre Manu Tuilagi replaced by Matt Smith and Horacio Agulla taking Smith's familiar wing slot.
Saracens, meanwhile, saw Neil de Kock start at scrum-half after winning a coin-toss for the role with England international Richard Wigglesworth, who was on bench duty.
But it was de Kock's opposite number Ben Youngs who made the first contribution, receiving a yellow card for illegally halting a Saracens attack that had de Kock at its forefront.
Farrell kicked the resulting penalty, and a tense opening quarter saw Saracens continue to look the more inventive team, piling pressure on a Tigers outfit that temporarily switched centre Anthony Allen to scrum-half duties.
A Flood penalty tied the scores with Youngs still off, and another strike apiece made it 6-6 before Saracens took control and enforced their territorial supremacy by claiming the opening try.
Quick lineout ball allowed Saracens freedom to attack, and hooker Schalk Brits roamed free before delivering a scoring pass to Short, who cut inside his opposite number Agulla.
Barnes required confirmation from video official Graham Hughes before awarding the try, but Farrell's touchline conversion confirmed Saracens' degree of control.
Leicester could not break free, and another Farrell penalty made it 16-6 before Flood completed his hat-trick, cutting the deficit to seven points after a half Saracens dominated.
Both goalkickers were at it again early in the second period, adding a further penalty apiece, before the first substitutions materialised with Wigglesworth replacing de Kock and Rhys Gill taking over from Matt Stevens.
Leicester also made two key switches, sending on players' player of the year Thomas Waldrom for number eight Jordan Crane and then England prop Dan Cole instead of Martin Castrogiovanni.
Saracens though, were in no mood to relinquish their chance of silverware, despite a frenzied finale when Leicester pushed and pushed for another league crown, but Saracens were not to be denied.
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