The Devon club, without a game since December 14, never recovered from Richardson's opener in the second minute, and Wearne's back post header five minutes into the second half all but sealed the points for John Dent's side.
The result leaves Town six points clear of both Tavistock and Cullompton Rangers near the bottom of the SWPL premier division.
That the game went ahead at all was something of a minor miracle.
At times during the first half there were fears the players would be unable to continue as spells of heavy rain turned the already-sodden Bickland Park pitch into a mudbath.
But despite the bad weather Falmouth played some of their best football this season.
Victory means another three points, and with it a first clean sheet at home. Dent's side remain unbeaten in 2014.
Richardson came back into the starting eleven after a spell out with a hamstring injury, and he announced his return almost immediately when his curled effort from the edge of the box saw Tavistock keeper Dean Smith go down early, and the ball bobble over his head and into the back of the net.
Harry Evans might have extended Town's lead just before the break when he slid a shot across the face of goal with Andy Westgarth lurking inside the six yard box.
But at 1-0 down Tavistock were still in the game, until five minutes after the restart when Wearne rose unchallenged at the back post to nod in Ed Timmons's corner.
Evans spurned another goal scoring opportunity mid-way through the half when he beat two defenders before seeing his poked effort hang up in the mud, and Smith scramble back to make a goal line clearance.
The Tavistock keeper made another good save from an Evans volley late on, and Westgarth saw his effort from just inside the box sail over the bar as Town finished strongly.
But 2-0 it remained.
Dent's side have now taken eight points from their last five games.
And despite the persistence of the clouds hanging over Falmouth there is at least now daylight between Town and the premier division drop zone.
Falmouth Town: D Burrows, E Timmons, D Batchelor, T Whipp (c), J Cooper, R Pope, D Richardson, G Bell, R Wearne, A Westgarth, H Evans. Subs (not used): L Dent, S Kellow, B Bassett.
Tavistock: D Smith, R Moorehead, H Southcott (S Cooke 81), D Hallett (c), J Ahearn, J Robins, J Phelpe, J Kelly, J Crag, I Rooke (N Ewing 78), A Davey (J Brewster 51). Subs (not used): J Cooke.