Scrum-half Takulua kicked two penalties to put Newcastle 6-0 up at half-time against a wasteful Worcester side.
A third penalty edged Falcons further ahead before Will Welch was sin-binned as Worcester grabbed a penalty try.
Wynand Olivier's score edged Worcester ahead, but Tongan scrum-half Takulua converted his own try to seal victory.
The win does enough to mathematically keep Falcons in contention for a place in European club rugby's top competition for the first time in 12 years, as the team finishing seventh in the table earn the chance to qualify through the play-offs.
However, a win for Northampton at Exeter on Saturday ends all hopes, which also come with other permutations outside their final league standing.
Falcons' aspirations were under real threat throughout at Kingston Park, with Worcester seeing a number of good chances go begging before the break, the best of which saw Perry Humphreys knock-on close to the line.
A penalty try from a maul, which also saw Newcastle's Welch yellow-carded, was their first response after going 9-0 down.
Olivier's try edged the visitors ahead for the first time, with both sides momentarily a man down soon after as Will Spencer was sin-binned for the hosts.
And as Worcester closed in for their first away league win for 13 months, Takulua dived over and kept his cool when adding the extra two points to cap a fine individual display.
Newcastle: Watson; Goneva, C Harris, Socino, Sinoti; Hodgson, Takulua; Vickers, Lawson, Welsh, Green, Robinson, M Wilson, Welch (capt), Chick.
Replacements: Cooper, B Harris, S Wilson, Fonua, Witty, Egerton, Willis, Tait.
Worcester Warriors: Pennell; Heem, Olivier, Te'o, Humphreys; Mills (capt), Hougaard; Rapava Ruskin, Taufete'e, Alo, Vui, Spencer, Potgieter, Lewis, Mama.
Replacements: Singleton, Bower, Milasinovich, Barry, Faosiliva, Baldwin, Lamb, Vuna.
Referee: Greg Garner
For the latest rugby union news follow @bbcrugbyunion on Twitter.
Sonatane Takulua scored all of Newcastle's points to beat Worcester, and maintain slim hopes of challenging for a European Champions Cup spot.