The loss was their first at Brickfields in 2015 and it leaves them five points behind second-from-bottom Moseley, who picked up a bonus point away at Nottingham.
Albion, after a poor first half, looked like they were going to sneak at least a bonus point themselves when they hit back and closed to within six points of their visitors with just over a minute to go. But they lost possession from the restart and Carnegie took advantage to score a fourth try through replacement Jon Clarke in injury-time to seal a five-point win.
Plymouth had gone into the match knowing they could possibly move off the foot of the Championship table for the first time since the start of the season.
But they really struggled to get going in the first 40 minutes and Carnegie, playing their first game under new coach Bryan Redpath, deservedly went in 18-3 up.
The home side were better after the interval but did not do enough to win.
They will be just glad Moseley only picked up a single point at Nottingham rather than a victory. However, the pressure is on Plymouth now to beat West Country rivals the Cornish Pirates away next weekend.
Albion, who had gone four games unbeaten at Brickfields, made three changes to their team that had lost at leaders Worcester Warriors in their last outing.
Influential back-row forwards Sam Simmonds and Iain Grieve joined Rob Langley and Herbie Stupple on the injury list, but Albion were able to welcome back player-coach Rhys Oakley, while Tom Jubb came in at flanker after impressing as a replacement at Sixways.
The other change was in the second-row where Canadian international Brett Beukeboom replaced dual-registered Tom Hicks.
The visitors, who included former Albion prop James Tideswell in their starting line-up, made a good start and dominated early possession.
Fly-half Harry Leonard had the chance to put his side in front on four minutes with a penalty but he hit the post and the ball fell kindly for Plymouth.
But four minutes later he did put Carnegie ahead with an easier effort.
The visitors were well on top in the early minutes and it was no surprise when they doubled their lead after a quarter of an hour with another Leonard penalty.
Albion were under a lot of pressure but somehow they managed to keep Carnegie out.
Christian Georgiou, who had rescued a draw for his side against Plymouth at Headingley in November, thought he had scored in the left corner on 24 minutes but it was ruled out by the touch-judge.
Albion struggled to get into Carnegie’s half but scrum-half Josh Davies did produce a good break after a tap penalty but the home side could not take advantage.
However, on 31 minutes they did cut the gap when Cusack kicked a penalty.
The Irishman had the chance to level the match a minute later with a 45-metre effort but this one was off target.
Yet, just when it look like Albion were getting back into the game, Carnegie, whose backs looked lively the whole match, found another gear.
Centre Peter Lucock was only denied by a great tackle by Ben Woods but a minute later the visitors did score the first try when they drove over from a line-out. Flanker Chris Walker was given the try, which was converted by Leonard to make it 13-3.
And it got worse for Albion on the stroke of half-time. Another good Carnegie attack saw Oakley sin-binned and the visitors too advantage to score again from a catch-and-drive move courtesy of Walker.
Leonard was unable to convert this time, but he team had an 18-3 half-time lead.
Albion came out a lot better after the break.
They didn’t conceded any more points while Oakley was off the field and when he was back they came agonisingly close to scoring a try. Many Albion fans thought they had scored on the posts but referee Darren Gamage only awarded them a five-metre scrum and the home side could not make it tell, with the official penalising them at the set-piece.
Plymouth just could not get any joy out of Gamage and somewhat against the run of play in the second half, Carnegie scored a third try on 62 minutes.
Fly-half Leonard was the man to score it but it was made by a storming run by prop Charlie Beech, who shrugged off three attempted tackles. Leonard could not convert but his side led 23-3.
Albion, with their replacements making an impact, gave themselves a lifeline when they were awarded penalty try after a good maul from a line-out. Cusack converted to make it 23-10.
The home side had two great chances five minutes later when they had two five-metre line-outs but they could not make either tell.
Marc Koteczky, who was influential after coming on at half-time, made a good break after collecting a clever Cusack chip but he was stopped 25 metres out.
Eventually, however, Albion’s pressure did tell when replacement winger Mark Kohler scored his first league try for the club in the right corner. Cusack brilliantly converted to put Albion in bonus point range.
But their hopes of snatching the late win or hanging on for point disappeared from the restart. They failed to keep hold of the ball and Carnegie attacked down the right with replacement Clarke going over in the corner to secure his team the win and the extra bonus point.
Plymouth Albion: 15 Lawrence Rayner, 14 Jack Arnott, 13 Ben Woods, 12 Toby Howley-Berridge, 11 Jake Henry, 10 Declan Cusack (capt), 9 Josh Davies, 1 Tom Fidler, 2 Ted Stagg, 3 Andrew Tiedemann, 4 Brett Beukeboom, 5 Neale Patrick, 6 Tom Jubb, 7 Eoghan Grace 8 Rhys Oakley.
Reps: 16 Tom Cowan-Dickie, 17 Dan Pullinger, 18 Luke Chapman, 19 Alistair Bone, 20 Marc Koteczky 21 Cameron Setter, 22 Mark Kohler.
Leeds 15 Christian Georgiou, 14 David Doherty, 13 Peter Lucock, 12 James Fitzpatrick, 11 Oli Goss, 10 Harry Leonard, 9 Chris Pilgrim, 1 Charlie Beech, 2 Phil Nilsen, 3 James Tideswell, 4 Matt Smith, 5 Chris Jones, 6 Jarad Williams, 7 Chris Walker, 8 Ryan Burrows (capt). Reps: 16 Joe Graham, 17 Lee Imiolek, 18 Ben Hooper, 19 Ben Harris, 20 Josh Bainbridge, 21 Sam Egerton, 22 Jon Clarke
Referee: Darren Gamage
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