Diego Jurd sent over the simple three from in front of the posts to split the teams after a keenly-contested clash, played out in foul conditions at a windswept Kingston Park.
The home side got off to a lively start, Alex Hearle getting them into the La Rochelle 22 with a super running line against the grain.
Ollie Leatherbarrow picked and went from the base of a five-metre scrum, and whilst the visitors’ defence held firm on both occasions, they were powerless to stop the boot of Brett Connon when the fly-half sent over a penalty from 30 metres out in front.
Stout defence maintained the Red Bulls’ lead on the quarter mark when La Rochelle kicked a penalty to the corner without return – Newcastle halting the maul at source and snuffing out the resulting phase play with aggressive line-speed.
The Frenchmen took the lead on 22 minutes, however, when the TMO awarded a try to Ihaia West, determining that the full-back had won a race to the bouncing ball from a grubber in behind the try-line.
La Rochelle doubled their score on the half-hour when a long, looping pass paved the way for winger Jules Favre to dive in at the right corner, and it got even better for the visitors when hooker Gabin Gabault scored from a trick play at the front of a line-out down the left.
They were reduced to 14 men when West was sin-binned for an off-the-ball tackle following Oli Spencer’s line-break, and from the resulting penalty Newcastle eventually conjured a try for Elliott Obatoyinbo – Connon finding his full-back with a looping pass for an unconverted try down the right.
That cut La Rochelle’s half-time lead to 15-8, but the Red Bulls drew level within two minutes of the resumption when replacement prop Murray McCallum made an instant impact.
The big Scot hit a rampaging running line from 20 yards out, stepping the full-back and racing between the posts, before Connon’s simple conversion drew the teams level.
His side continued the momentum with a string of attacking forays, but it was the boot of Connon which put them back into the lead on 55 minutes when the fly-half sent over a penalty from just right of centre.
Newcastle were reduced to 14 men just before the hour mark when Fregus Lee-Warner was sin-binned for a high tackle, and the teams were level on 63 minutes when Diego Jurd slotted a penalty from 35 metres out on the right.
The fly-half hit the post with a long-range effort on a night of high drama, but made no mistake three minutes from time when his effort from in front of the posts sent La Rochelle on their way to the quarter-finals – Nolhann Couillaud removing any doubt with a try from the very last play.
Newcastle Red Bulls: 15 Elliott Obatoyinbo, 14 Christian Wade, 13 Alex Hearle, 12 Rhys Beeckmans, 11 Oli Spencer (Ben Healy, 56), 10 Brett Connon, 9 Simón Benítez Cruz (James Elliott, 51); 1 Adam Brocklebank (Micky Rewcastle, 68), 2 George McGuigan (captain, Samson Adejimi, 58), 3 Richard Palframan (Murray McCallum, 40), 4 Jamie Hodgson (Tim Cardall, 56), 5 Freddie Clarke (Fergus Lee-Warner, 51), 6 Tom Gordon (Ollie Leatherbarrow, 69), 7 Tom Christie, 8 Ollie Leatherbarrow (Reuben Parsons, 68).
Scorers - Tries: Elliott Obatoyinbo, Murray McCallum. Conversions: Brett Connon. Penalties: Brett Connon 2.
Stade Rochelais: 15 Ihaia West, 14 Jules Favre, 13 UJ Seuteni (Sacha Elissalde, 60), 12 Adrien Seguret, 11 Hoani Bosmorin, 10 Diego Jurd, 9 Thomas Berjon (captain, Nolhann Couillaud, 69), 1 Louis Penverne (Christian Luaki, 60), 2 Gabin Garault (Tolu Latu, 54), 3 Karl Sorin, 4 Thomas Lavault, 5 Simon Huchet (Aleksandre Kuntelia, 48), 6 Edouard Richer, 7 Lucas Andjisseramatchi, 8 Watisoni Waqanisaravi (Judicael Cancoriet, 47).
Scorers - Tries: Ihaia West, Jules Favre, Gabin Garault, Nolhann Couillaud. Penalties: Diego Jurd 2.
Referee: Eoghan Cross (Ireland). Attendance: 4,274.