In a disjointed first half, neither side could produce a clear opening as Jose Fonte and Rickie Lambert went closest for Saints.
The game finally came alive on the stroke of half-time when Adam Lallana was denied a penalty after being hauled down as he stormed in Brighton's box, while Gary Dicker saw a deflected shot go close at the other end.
The second half was similarly hard fought but Brighton spurned a golden chance to snatch all three points on 66 minutes.
Ryan Dickson was adjudged to have fouled Glenn Murray in the Saints area, but West Brom loanee Wood - who scored from the spot last week against Bristol Rovers - saw his well-struck penalty saved by Davis.
Despite a flurry of late end-to-end action, chances remained scarce as the match ended all-square, keeping the Seagulls on top of League One and Saints lurking in the play-off picture.