The former England fly-half, who went into the match on 1,992 points, achieved the milestone with his third penalty in the 27th minute.
England winger Chris Ashton marked his Saracens debut with two second-half tries - after being sin-binned in the first half - with England centre Brad Barritt and replacement Nils Mordt also touching down.
Hodgson finished with 20 points, adding four conversions to his three early penalties, with full-back Alex Goode also landing a penalty.
The only Irish reply was a penalty from full-back Tom Homer, last season's Premiership leading scorer with 278 points.
After Homer missed his first penalty shot in the second minute, Saracens skipper Steve Borthwick had a fourth-minute try ruled out.
Borthwick was denied by the new regulations which allow the referee to refer the build-up to a try to the television match official - instead of just seeking advice on the act of scoring.
Hodgson created the opening before man-of-the-match prop Mako Vunipola turned the ball inside to send Borthwick over between the posts but TMO Trevor Fisher ruled that the final pass was forward.
After releasing another pass, Hodgson had to survive a bone-shaking late challenge from Ofisa Treviranus which earned the Samoan flanker a 15th-minute yellow card.
While Hodgson recovered, Goode landed the resulting penalty.
Saracens were also reduced to 14 men when Ashton, a summer capture from Northampton, was sin-binned by referee Wayne Barnes for a late barge on Irish fly-half Steve Shingler.
Homer, who had missed three early long-range penalties, finally succeeded at the fourth attempt, from just inside the Saracens' half, to open the Exiles' account with the resulting kick.
Hodgson was not to be denied, however, and he landed his second penalty, in off the post, in the 25th minute.
Two minutes later Hodgson, who scored most of his points for former club Sale Sharks, took his Premiership tally to 2,001 with his third penalty of the match to give his side a 12-3 lead, which they retained until the end of a feisty first half.
Ashton made amends for his earlier misdemeanour by scoring the first try of the match with the second half only two minutes old.
Hooker Schalk Brits, former Wigan Warriors rugby league star Joel Tomkins and USA international winger Chris Wyles were involved in a quality build-up which ended with Ashton, up in support and showing his best predatory instincts, supplying the finish despite a despairing attempt at a tackle from new Exiles skipper Bryn Evans.
Hodgson converted to stretch the Saracens' lead to 19-3.
Eight minutes later Hodgson was on target once more after Saracens struck again with a try from England centre Brad Barritt.
But Irish looked unfortunate not to have the score referred to the TMO for an apparent forward pass in the build-up before Ashton knocked the ball back, one-handed, for Barritt to pounce.
Saracens flanker Kelly Brown was the third player to be sin-binned, for killing the ball at the breakdown midway through the second half, but his 14 team-mates kept their line intact until he returned.
Back to full strength, Saracens had the final say with two more tries, both converted by Hodgson.
The fly-half passed out of the tackle to set up Ashton for his second try in the 73rd minute and Mordt secured the bonus point with their fourth three minutes from time.
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