The on-loan Southampton shot-stopper kept out a 61st minute spot-kick by Derby's Spanish midfielder Inigo Idiakez with his feet as he dived to his right.
Blayney's stop spared the blushes of Brighton defender Gary Hart, who conceded the penalty by fouling Marc Edworthy.
It was a let-off for Hart following his controversial late red card in the 3-2 home defeat by Crystal Palace the previous Sunday.
The penalty incident was one of the few highlights of a dour encounter between two teams labouring in the wrong half of the table.
It was surprising to see Idiakez place the penalty-kick too close to the centre of the goal. He is a renowned set-piece specialist and had earlier twice gone close to giving Derby the lead.
His in-swinging corner in the 17th minute curled over Blayney and had to be headed off the line at the far post by Brighton full-back Paul Reid.
Idiakez also went close with a free-kick into the side netting from 25 yards shortly before his spot-kick blunder.
Chances were few and far between for next-to-bottom Brighton, who have now won only once in their last 16 home matches.
A dangerous cross in the first half by midfielder Dean Hammond eluded his team-mates inside the Derby box in a rare foray forward.
The south coast club almost snatched the points after manager Mark McGhee made a double substitution.
Jake Robinson, one of the replacements, exploded a 35-yard shot against the crossbar seven minutes from the end.