An interception try from right wing Richard Haughton plus a try, two conversions and a penalty from fly-half Derick Hougaard gave Brendan Venter's side a 17-15 half-time lead, with left wing Chris Wyles crossing late on to seal the win.
Newcastle's points came courtesy of five penalties from the boot of fly-half Jimmy Gopperth during a pulsating first half but they were unable to make the most of the chances they created and they remain winless this season.
Newcastle's bid for their first win of the season, having drawn three of their first four games, got off to the worst possible start as scrum-half Micky Young's pass was picked off by Haughton, making his first start of the season, who strolled under the posts from 30 metres to give Hougaard the simplest of conversions after only five minutes.
Fly-half Gopperth immediately got the Falcons on the board with two penalties after some enterprising attacking play, Hougaard replying with a penalty of his own to put Sarries 10-6 up.
Newcastle sought to keep the ball in hand and deny the visitors the chance to employ the effective kicking game that had seen them make an unbeaten start to the season, and it proved effective as Saracens continued to give away a stream of penalties, Gopperth gratefully landing another three kicks to give the hosts the lead for the first time.
It took Saracens until the 33rd minute to produce a string of attacking phases after a set move allowed van Heerden to make good yards in midfield. The ball was recycled quickly only for wing Chris Wyles to drop the ball with the line at his mercy. But the Falcons were unable to clear from the resulting scrum and Hougaard dummied his way over wide out for a converted try, giving the Vicarage Road side a 17-15 lead at the break.
Saracens' dominance finally told when, having spent a prolonged period of the game in the Newcastle 22, Fijian centre Kameli Ratuvou got his hands out of an attempted tackle to allow Wyles to cross with four minutes to play and atone for his earlier error.