Flanker Dombrandt scored from an interception late in the first half in a game where defences came out on top, while Marcus Smith chipped in with two penalties and a conversion.
Replacement Saili finished off a Joe Marchant break midway through the second half, while Andrew Davidson scored the Falcons' only try minutes after.
Dean Richards' men have now lost six of seven matches in the Premiership this season, while Harlequins move above Leicester Tigers and into fifth.
Newcastle had rediscovered their form during the European and Premiership Cup fixtures but they got off to a bad start at the Stoop, with fly-half Joel Hodgson carried off with an injury inside ten minutes.
Harlequins fly-half Smith got the first points of the night for the home side, kicking a simple penalty from close range.
However, they suffered problems at the set-piece and hooker Elia Elia was substituted before half-time after some wayward throws at the line-out, while the scrum proved to be a reliable source of penalties for the Falcons.
But the visitors also struggled to create going forwards and Toby Flood missed their best chance of points in the first half, leaving a penalty fractionally short.
They paid for it minutes later when flanker Dombrandt picked off a pass in midfield and scampered through unopposed to score under the posts, Smith adding the conversion for a 10-point half-time lead.
The Falcons increased the tempo after the break but Harlequins remained defensively vigilant and resilient, winning a crucial penalty inside their own 22 just as the Falcons were gaining a head of steam.
Smith was the next to score, booting his second penalty of the match through the posts from 30 metres out.
The Falcons continued to play with a high intensity but the home side were too efficient in defence and went on to score the second try which killed off the match.
Wing Nathan Earle and centre Marchant combined superbly down the left, with the latter sidestepping one tackle and bursting for the line.
He was tripped just short but popped up a pass to the supporting Saili and the centre touched down.
That woke the Falcons up and they responded immediately, with lock Davidson scampering through from close range for their first points of the match.
But they could not make further inroads and the hosts held on.