In a cracking match, the English Championship side only secured victory in the last few minutes thanks to tries from full-back Grant Pointer and back-row Blair Cowan.
The Pirates, who are the holders of the trophy, were always ahead in the match but Currie kept going at them all afternoon and in Edinburgh back-up man Tom Brown had one of the best players on the pitch.
Pirates' player Dave Ward was commentating on the match for a local radio station and was impressed by Ally Donaldson's men. He said: "Currie's forwards were very competitive at the breakdown and, in the backs I was impressed by Tom Brown, Matt Scott and Hayden Abercrombie. They never gave up."
Despite these plaudits from the pool A loss, coach Donaldson will not have been happy with the way his side coughed up scores at key points of the encounter.
For example, a minute after Kiwi Abercrombie had kicked his second penalty to cut the deficit to 10-6 in the first quarter the Pirates went up to the other end and grabbed a try.
And, with Currie looking for a good start to the second half, they got anything but when Welsh winger Rhodri McAtee got his first try of the day just 47 seconds after the restart to make it 28-14.
In the first half, Currie had bounced back well from losing a try after just six minutes and kept themselves in the game thanks to a try from second-row Ryan Wilson and the boot of Abercrombie.
In the second half, the Pirates, unbeaten at home this season, scored five tries - two from McAtee - but Brown and Matt Scott went over for Currie to keep things interesting. Winger Alex Dunbar also went close on a couple of occasions.
Currie now take on Worcester Warriors this weekend in the same event.
Scorers: Cornish Pirates: Tries: Davis, Elloway, Hopper, McAtee 2, Bentley, Pointer, Cowan. Pens: Cook 2. Cons: Cook 2, Bentley 2. Currie: Tries: Wilson, M Scott, Brown. Pens: Abercrombie 4. Cons: Abercrombie 2.
Cornish Pirates: G Pointer; W Davies, D Locke, M Hooper, R McAtee; R Cook, J Doherty; P Andrew, R Elloway, J Currie, M Myerscough, I Nimmo, B Cowan, K Marriott, T Holmes. Subs used: C Rimmer, T Cowan-Dickie, A Paver, B Gulliver, C Morgan, S Winn, J Bentley.
Currie: T Brown; A Dunbar, H Abercrombie, A MacMahon, A Turnbull; M Scott, R Snedden; J Cox, S Simonsen, A Hamilton, R Wilson, G Temple, M Cairns, R Weston, M Entwhistle. Subs used: F Scott, R Merrilees, A Adam, A Best, A Binikos, J Smith, A Whittingham.
Referee: L Pearce