The Clarets started the afternoon in 19th place, without a win in eight matches, whilst the vultures were swirling around manager Sean Dyche just seven months after he guided his Burnley side into the Europa League qualification play-offs.
And the nerves were apparent in a lacklustre start to the contest with numerous passes going astray on both sides of the pitch, despite Brighton's positive start to the campaign.
The visitors gradually started to build some control with various spells of possession but failed to test Joe Hart who was making his 400th league appearance in English football. Yves Bissouma dragged wide from range but that was as good as it got in the opening 20 minutes.
Burnley managed to get a foothold in the game through their creative Iceland international, Johann Berg Gudmundsson. The wide player, who has created four goals already this season, began to manufacture some devilish crosses and Robbie Brady volleyed over after connecting with one on the 30-minute mark.
Finally, some goalmouth action ignited the contest several minutes later. Brighton goalkeeper Mat Ryan was forced into two smart stops as both Chris Wood and Brady were denied in quick succession. The hosts were building in confidence.
Moments after a surging run from Anthony Knockaert resulted in a strike that flew over the crossbar in a brief spell of respite for the visitors, the home side broke the deadlock.
A game of pinball emerged in the away penalty area as Brighton failed to clear their lines and Jack Cork's strike was eventually deflected home by Tarkowski. A scrappy goal to say the least but one that gave Burnley a crucial lead at the break.
Brighton looked slightly stronger after the break but Burnley responded by tightening up defensively. In fact, the visitors didn't cause Hart any problems until Ben Mee's wayward clearance from a Davy Propper cross flew into the arms of his own goalkeeper.
Tarkowski went close to doubling the advantage moments later as he headed a Cork corner wide before Brighton started to regain control of possession.
The visitors probed but again struggled to unlock a Burnley defence that were beginning to gel as they did last season. Solly March started to have more of an influence as his cross was headed wide by Lewis Dunk before the midfielder smashed an effort off-target himself.
Burnley of old
Possibly the best opportunity of the contest for the visitors fell to Jurgen Locadia with less than ten minutes remaining. He found himself in space to meet a Glenn Murray cross but could only head over.
March dragged another effort wide late on but Burnley held firm in a desperate spell of injury time defending to earn a valuable three points.
A vital win that lifts the Clarets two points clear of the drop zone.