Backed by their vocal support, Bulgaria, who were unbeaten in seven matches heading into the game, should have taken the lead in the fourth minute after a chaotic goalmouth scramble.
Georgi Kostadinov forced Peacock-Farrell into a superb save from a corner and Kolev powered the rebound towards goal. Ciaron Brown had to scramble the forward's effort off the line but his clearance smashed off Aleks Petkov, who inadvertently hit the upright before Peacock-Farrell was able to smother the danger.
Despite half-chances for Isaac Price, Daniel Ballard and Conor Bradley, it was a sign of things to come.
Moments after firing wide from range, the impressive Despodov fired in a free-kick that rattled the post and bounced out of play.
PAOK midfielder Despodov was dictating play and his stunning through ball almost set up Kolev, but the forward's lobbed finish lacked the power to trouble Peacock-Farrell.
Northern Ireland were almost gifted the opener when Aleks Petkov misjudged Ballard's long ball, which allowed Isaac Price through on goal but he rushed his effort, which flew well wide.
But, in contrast, Bulgaria made no mistake when handed a chance of their own as Peacock-Farrell's pass caught Ballard on his toes inside the area, which allowed Kolev to race in and square the ball for Despodov to tap home at the back post.
Ballard had a great chance to equalise in the final play of the half when he headed over Bradley's cross.
Northern Ireland started the second half on the front foot but struggled to create many clear-cut chances.
Dion Charles saw an attempt well blocked just shy of the hour mark and substitute Shayne Lavery fired straight at Dimitar Mitov before being adjudged to have been offside.
Lavery had another sight of goal with three minutes to play but he headed Jamal Lewis' cross over the bar.
Just like in the first half, Ballard had a header on goal deep into injury-time but it sailed wide as Northern Ireland's hopes of rescuing a point ended.
Northern Ireland are back in action away to Belarus on 12 October before Bulgaria travel to Belfast three days later.
Image source, BBC Sport
