The 9-5 final score in favour of the Cornish side was as tense at the end as the slender margin suggests.
However, Albion's late push after Luke Cowan-Dickie's second-half try was a chance they should never have been given following a dominant opening half from Pirates.
The visitors' points all came from the boot of their fly-half – and former Albion captain – Kieran Hallett. They should have scored more, having spent virtually the entire first half deep in Albion territory.
After a series of scrums on the edge of the hosts' 22 in the opening ten minutes, scrum-half Gavin Cattle threw a missed pass to centre Tom Riley who knocked on when taking the ball into contact.
Albion, having seen off the early pressure struggled to build momentum of their own as their line-out constantly failed find its intended man.
It took Pirates 18 minutes to turn their superiority into points. It came through a Hallett penalty after prop Alan Paver was illegally brought to ground as he headed towards the Albion 22.
The former Albion ten then doubled his side's advantage four minutes later.
Pirates were pragmatic and solid. They played the conditions well, with foreknowledge of the forward-orientated game plan coming in the opposite direction.
Albion, meanwhile, were dogged in keeping their try line intact but failed to break into the Pirates' 22 with possession in hand until the second half.
Their best chance to reduce to deficit before the referee signalled the half-time break came and went when Paul Roberts' long-range penalty was sliced just wide of the right post.
A better start to the second half from Albion brought another chance for the home fly-half but, again, he was unable to find accuracy with his kick.
Albion's improved work rate continued and a number of pick-and-go raids for the line were repelled by Pirates at the cost of a penalty for offside, which Roberts again kicked wide of the posts.
The hosts were initially made to pay when Hallett slotted his third penalty of the game, but Albion came back strongly and set up the final 30 minutes when front-row Cowan-Dickie crashed over to the right of the posts for an unconverted score.
A second chance came when a chase into the corner forced Pirates full-back Wes Davies to rush his clearance kick into the touch. However, the resulting catch-and-drive attempt was knocked on. Hallett was able to claim back some territory for his side which they never relinquished again.
