The Falcons, who finished bottom following the 22-game regular Premiership campaign, will only be relegated if the eventual Championship winners meet top-flight entry criteria.
Bristol were one of those clubs, but a 29-18 second leg semi-final victory over Cornish Pirates was not enough to keep them in title contention.
Pirates' 45-24 first-leg triumph underpinned an aggregate success of 63-53, and they will now meet London Welsh in the two-legged final on May 23 and 30.
The Pirates did not put themselves forward for the Rugby Football Union's Premiership entry audit, and neither did Bedford, who were edged out 30-27 by London Welsh over two legs.
Bedford won Sunday's second leg 24-17 at Old Deer Park, but they came up just short after losing the home tie 13-3.
The Richmond-based Exiles are adamant they will meet Premiership entry criteria, which they underlined through a statement released earlier this week by club chairman Bleddyn Phillips and the board of directors.
It is thought that they are considering a possible ground-share with League Two football club Oxford at the Kassam Stadium, although a move to Harlequins' Twickenham Stoop has also been mooted.
The RFU and Premiership Rugby intend clarifying the promotion situation ahead of the Championship final, yet Newcastle are still clinging to survival.
Dean Richards, whose three-year worldwide coaching ban expires this summer following his role in the Harlequins' Bloodgate saga, will take charge of the Falcons next season.
Despite beating fellow strugglers Wasps away from home last weekend, Newcastle still ended propping up the table by a point, and they will now hope that the Pirates emerge victorious after the final's second leg.