A brave performance in a thrilling match saw the Reds claim four of the game's nine tries in claiming two points in a brave defeat to unbeaten hosts Yorkshire Carnegie.
A helter-skelter match saw Jersey take an early lead before battling to stay in touch during the final hour, with the hosts having been 19-5 up within the first quarter.
The Reds made a great start with a penalty almost from kick-off that Cope kicked to the corner. A catch-and-drive move was finished by Jack Macfarlane - his first league try for Jersey - and the visitors led 0-5.
The lead was short-lived, with the home side spinning the ball wide after a series of mauls and Andy Forsyth crossing in the corner, with Joe Ford adding an impressive conversion.
Back came the Reds with a Cope penalty to retake the lead, but then Carnegie turned up the power with Seb Stegmann and Ollie Stedman crossing and helping their side to a 19-8 lead.
Ford added a penalty after a spell of pressure, but the Reds had a strong finish to the half: Ross Adair could have scored had he held a Joel Dudley pass, but soon afterwards Cope crossed from a sublime George Eastwell offload, and added the extras to make it 22-15. The half-time whistle blew with the hosts taking heart, especially with Carnegie prop Lewis Boyce recently despatched to the sin-bin for persistent offence.
A Jersey mistake from the restart gave Ford the chance to add a straightforward penalty (25-15). This sparked the Reds into action, determined to take advantage of their extra man, and after a couple of missed opportunities the ball was spun wide and Adair streaked over, with Cope adding a magnificent conversion (25-22). The sin-bin period had been productive for the Reds, who scored 14 points to 3 in this spell.
Tries by Forsyth - his second - and Ford helped the hosts pull clear, but a brace of missed conversions by Ford meant the Tykes (as they used to be called a few rebrands ago) weren't out of sight.
Macfarlane claimed the Reds' fourth try and Cope's conversion narrowed the deficit to just six points. This left the visitors holding two bonus points - in the final throes they could have gained three more, but were penalised for crossing, or lost one - when Ford went for goal. But the last-kick penalty went wide and both sides had cause for satisfaction: Carnegie maintaining their unbeaten record, and the Reds showing great heart in taking points from a team with Premiership aspirations.
Head Coach Harvey Biljon said afterwards: "We've had reduced numbers but the boys were prepared to put an effort like that in - I don't know how they did it, but we've scored 11 tries in two games and I hope the wheel has started to turn - that's something we need to build on.
"We were absolutely going for the win - two points at Headingley is good but our errors gave them opportunities and Carnegie are too good a side not to take them.
"We knew it was important to score when they were a man down; we gave away a silly three but overall did well. London Scottish was a low point, but we have got ourselves back on track - now we have to back it up at home next week against Pirates."
YORKSHIRE CARNEGIE: McColl (Beech 45-52); Holmes, Forsyth (Wright 75), Lucock, Stegmann; Ford, Davies (Green 70); Boyce (YC 39, Beech 78), Mayhew (Nilsen 61), O’Donnell (Cusack 56), Schofield (Whetton 52), Smith, Burrows, Beck (Saull 75), Stedman.
JERSEY REDS: Adair; Watkins, King (Katz 65), Eastwell, Davies; Cope, Dudley (Hardy 50); Thomas, Macfarlane, Kerrod (Armstrong 70), McKern, Voss (Kolo'ofa'i 56), Rae (Argyle 65), Freeman (c), Haining. Replacements (unused): McCarthy, Woolmore
REFEREE: Tim Wigglesworth (RFU); Assistants: Wayne Falla and Greg Macdonald
Half Time: 22-15
Tries: Forsyth 4, 58, Forsyth 14, Stedman 19, Ford 65
Conversions: Ford 4, 19
Penalties: Ford 30, 42
Tries: Macfarlane 2, 70, Cope 40, Adair 49
Conversions: Cope 40, 49, 70
Penalty: Cope 11
Why not get yourself a Hot Pot (Curry or Chilli served with rice) in the supporters marquee £8 per pot.