Conceding four first-half tries left the Reds too much to do in their final B&I pool match, but there's still a home quarter-final back on the Island to look forward to at the end of March.
Brendan Cope scored all the visitors' points and helped the Reds edge back into contention, but the hosts held on for the win.
The game at Silver Royd, home of Scarborough RUFC, had been in doubt due to the adverse weather of the previous few days, but the pitch was passed fit after a morning inspection.
A lively start saw Chris Elder storm over for the hosts in the second minute, but Brendan Cope hit back for the Reds with a try, conversion and penalty.
The key period of the game was the closing 12 minutes of the first half, when Carnegie added three tries to their tally. Elder set up Stevie McColl during a lightning counter attack and then the full-back claimed his second. Just before the break Harry Newborn was on the end of a well-executed rolling maul and all the Reds could manage in return was a second Cope penalty.
In spite of having at least an equal share of possession in the second half, Carnegie couldn't add to their points total, while Cope added two more penalties to reduce the arrears to seven. The game became feisty, with Matt Smith and James Voss earning yellow cards after an altercation, but the home side survived with the Reds unable to build a platform for what could have been a tying score.
There were contrasting emotions for the two sides an hour after the game as the full-time result from the Leinster 'A' v Doncaster Knights' game came in. The Knights scored 28 second half points to win 28-37 and qualify ahead of Carnegie, who must now await the result of the postponed Cornish Pirates v Hartpury game to see if they'll qualify. Meanwhile the Irish hosts landed just a single bonus point that left the Reds second, behind Ealing, in the final pool-stage rankings.
The four top-ranked qualifiers, who'll enjoy home advantage in the quarter-finals, are Ealing, the Reds, Leinster 'A' and Bedford Blues. The other qualifiers, in no particular order, are Doncaster, Ulster 'A', Munster 'A' and either the Pirates or Carnegie.
The Reds will play their home quarter-final against opponents TBC on the Friday or Saturday of the Easter weekend (March 30th/31st).
YORKSHIRE CARNEGIE: Elder; McColl, Burdon, Bullough (Forsyth 76), Brown; Lucock (Irvine 59), Homer; Vaughan (Thraves 46), Newborn (Poole 71), Beech (Vaughan 71), Myerscough, Smith (YC 55), Bainbridge (Lloyd 79), Mayhew (c), Steadman (Whetton 74).
JERSEY REDS: Cope; Pincus, Van Breda (c), Eastwell, Worrall; Bryant, Hardy (Duchesne 78); Godfrey (c) (Woolmore 73), Macfarlane (Selway 78), Atalifo (Clarey 67), Voss (YC 55) (Quarrie 65), Mckern, Bartle, Nagle-Taylor, Upfield. Unused replacements: Joyce, Best.
REFEREE: Jack Makepeace
Half Time: 26-13
Tries: Elder 2, 32, McColl 28, Newborn 39
Conversions: Homer 28, 32, 39
Try: Cope 6
Conversion: Cope 6
Penalties: Cope 22, 30, 48, 63
Why not get yourself a Hot Pot (Curry or Chilli served with rice) in the supporters marquee £8 per pot.