Dimitar Rangelov's early header gave the hosts the lead at the Vasil Levski National Stadium, but Nicklas Bendtner got the visitors back on level terms soon after Ivan Bandalovski's 27th-minute red card.
The result left the Danes still searching for their first win of the qualifying campaign, having drawn 0-0 with the Czech Republic in their first match last month. Bulgaria, meanwhile, slipped two points off the pace following group leaders Italy's 3-1 win in Armenia earlier this evening.
Rangelov headed Bulgaria into an early lead, connecting with Georgi Milanov's cross from the right in the seventh minute.
The hosts suffered a setback, though, when CSKA defender Ivan Bandalovski was sent off 20 minutes later for a bad foul on Daniel Wass. And Denmark made their extra man tell before half-time, Juventus' on-loan Arsenal striker Bendtner slamming his header into the top-right corner following Michael Krohn-Dehli's cross from the left.
Denmark's Christian Eriksen went closest to grabbing a winner after the interval, the Ajax midfielder unleashing a left-footed attempt in the 51st minute only to see it hit the bar.