Andrews lasted almost three quarters of the match after joining from Natal Sharks, while fellow new recruit and countryman Warren Britz did not figure after also arriving recently.
Two early penalties from England fly-half Jonny Wilkinson got the home side off to a good start on the day that they officially opened their newly-refurbished stadium, but they went behind when hooker Tani Fuga crossed for a Harlequins try, converted by Paul Burke, who had earlier kicked a penalty.
Almost straightaway Newcastle answered back through Andrews' second row partner Stuart Grimes, who scored a try which Wilkinson duly converted.
Back rower Hugh Vyvyan was next over for the Falcons five minutes after the interval, Wilkinson's conversion seeing the troubled side stretch away from the visitors in front of a vocal crowd in freezing conditions.
Centre Jamie Noon added to that eight minutes later with their third, and a valuable bonus point was secured when England fullback Michael Stephenson crossed the line with seven minutes to go – Fuga grabbing a consolation for Quins in the last minute.
For Newcastle Falcons:Tries: Grimes, Vyvyan, Noon, StephensonCons: Wilkinson 3Pen: Wilkinson 2
For NEC Harlequins:Tries: Fuga 2Con: BurkePen: Burke