The New Zealander booted all 18 of his side's points in a near-perfect kicking display at Goldington Road.
The hosts Bedford replied with penalties from Myles Dorrian (two) and replacement Jake Sharp.
Bedford actually drew first blood when Dorrian slotted over a sixth-minute penalty but Newcastle, who had won all three of their previous meetings between the two sides this season, fought back to lead 12-3 at half-time.
All of those points came from the boot of Gopperth, who confidently fired over penalties in the 12th, 16th and 28th minutes before adding another right on the stoke of half-time.
Gopperth was leading Newcastle's charge and he looked set to score the opening try of the match immediately after the restart following a rapid Falcons break but he was dragged down by a crucial tackle from James Short with the line in his sights.
Gopperth was then penalised for getting up and trying to play on after the tackle, but Newcastle quickly won possession back and following good work by their forwards won another penalty in front of the posts. The visiting fly-half was never going to miss from that range and he duly slotted over his fifth penalty of the night to make it 15-3.
It was soon 18-3 thanks to another Gopperth penalty before Bedford tried to launch a fightback which saw them reduce the deficit by three through Dorrian just before the hour. Gopperth had a chance to extend Newcastle's lead back to 15 points again with nine minutes to go but for once his radar was off as his penalty drifted wide.
With the tie slipping away, Bedford pushed hard to try to cut the gap in the closing stages and they gained some reward for a sustained spell of pressure when replacement Sharp fired over a penalty after the home side opted not to go for the corner.
Sharp also had another chance right at the end but he pulled his kick from wide out left and Dean Richards' Newcastle, who lost just one match during the regular season, will now hope to finish the tie off at Kingston Park and seal their berth back in the top flight.
- Sports & Recreation
- Myles Dorrian