Worcester had opened the scoring with a Joe Carlisle penalty after five minutes when Ally Hogg strayed offside but Hogg made amends with a pulverising drive which Jamie Helleur and Crone Uys carried on to win a penalty which Gopperth knocked over for 3-3 after 12 minutes.
Carlisle's high kick was taken by Greg Goosen but the Warriors pack turned it over and Newcastle were offside again for Carlisle to put Worcester back in front at 6-3 in the 20th minute. But Worcester conceded a soft penalty at the line-out two minutes later and Gopperth levelled the scores.
Newcastle were struggling to find any distance with their kicks and an appalling slice from Rikki Sheriffe allowed Worcester to drive into the Falcons 22 and Shaun Perry looped around Carlisle with the fly-half offloading in the tackle for Perry to score, and Carlisle's conversion in the 27th minute put the Warriors 13-6 ahead.
Newcastle tore into Worcester early in the second half to gain a penalty after seven minutes which Gopperth kicked for 13-9 to give the Falcons hope. Alex Tait scythed through the Warriors defence and a try seemed certain but Uys just failed to hold the pass which Tait popped up as he was hauled down inside the 22.
With 12 minutes left Andy Goode kicked a 45-metre penalty which looked to be the final act until that dramatic match-saving effort from the Falcons.