Tom Saintfiet's men had control for most of the first half but their profligacy in front of the goal allowed Bhutan to settle for a draw at the Bangabandhu National Stadium on Tuesday evening.
Bangladesh broke down the right in the 16th minute and Rubel Mia whipped in a low cross but Shakhawat Hossain Rony failed to tap the ball in from edge of the six-yard box.
Moments later, another Rony header on another Rubel cross was brilliantly saved by Bhutan goalkeeper Hari Gurung.
Rony squandered yet another good opportunity five minutes before the break when he drove powerfully straight to the goalkeeper from inside the box.
Bhutan, ranked 192nd by FIFA, looked more organised after the break while 183-ranked Bangladesh lost their rhythm.
Bhutan's Tsherining Wangdi curled one from outside the box narrowly wide of the far post as the players looked weary on the muddy uneven pitch.
But Bangladesh made a mess of three straightforward opportunities in the stoppage time with Nabib Newaz Jibon wasting the easiest of them.
He found himself inside the six-yard box with only the goalkeeper to beat but could only manage a weak header straight at the keeper.
The failure of the strikers will weigh heavily on the team which suffered a 5-0 rout by the Maldives away in Male in a friendly last week.
The second leg of the Asian Cup playoff will be held in Thimpu on Oct 10.