Victory ended a run of three consecutive league defeats for reigning champions Leicester as they inflicted just the second Premiership loss suffered by Saracens this season.
The Tigers' triumph left them a mere three points behind fourth-placed Gloucester and kept them in the hunt for a place in the title-deciding playoffs.
Leicester surged into a 24-3 lead after 35 minutes to the delight of the Midlands' club's fans at Welford Road.
Pollard, Cameron Henderson and Mike Brown scored Leicester's tries, with Pollard, a key figure in the Springboks' 2019 World Cup final-winning team, converted all three scores and added a penalty for good measure.
Eroni Mawi and Alex Lewington crossed for Saracens, with former England international Alex Goode kicking two penalties and a conversion.
Goode led out Saracens to become the first player to reach 350 appearances for the London club in the professional era, with ex-England full-back Brown making his Leicester debut.
It took Leicester just six minutes to break the deadlock. On halfway, Jasper Wiese surged through a huge gap in the visitors' defence to run 55 metres before being hauled down inches short of Saracens' line. But the ball was recycled quickly and Pollard crashed over.
It was a a similar story eight minutes later as another 55-metre break led to a try.
Brown made the initial thrust on this occasion, with Tigers captain Julian Montoya, the Argentina hooker, twice involved in the movement before Henderson dived over.
Goode got Saracens on the scoreboard with a penalty before fine build-up flay from fellow veteran backs Jimmy Gopperth and Chris Ashton paved the way for Brown to evade the last defender for another well-worked try.
Saracens reduced the deficit when Mawi went over from close range for a try converted by Goode that left Leicester 24-10 ahead at half-time.
Goode kicked a penalty before Lewington pounced on a Leicester handling error to regather the loose ball and run in a try from 70 metres.
Saracens laid siege to Leicester's line but brilliant last-ditch defence kept them at bay as the hosts held out for a hard-earned win.
