Gloucester saw the majority of the ball in the opening 20 minutes, but had to settle for penalty kicks, with Twelvetrees booting the ball through four times.
The visitors got their opening try on the board when Willi Heinz played the ball out wide to Matt Kvesic, setting the flanker up for a short dash across the line.
Charlie Sharples was over just three minutes later, with Bill Meakes setting the winger up for a sprint over as Gloucester started to run away with proceedings.
Newcastle started to see more Gloucester territory in the second half, allowing Simon Hammersley to close the gap shortly after the restart, before Chris Harris broke free to race over for a second Falcons try.
Twelvetrees continued to put in a clinical performance with the boot for the visitors, adding nine penalties and a conversion with five minutes remaining, but a late try from George McGuigan gave Newcastle hope, but they were unable to find a fourth try in the final moments, allowing Gloucester to hold on for the win.