Left-arm spinners Price and Dabengwa returned with career-best bowling figure of 4-22 and 3-15, as Bangladesh got off to a slow start after winning the toss and electing to bat.
A damp pitch and murky overhead conditions provided Zimbabwe’s pacers with lots of movement in the early going, making it difficult for the Bangladesh batsmen to score.
Their first run from the bat came from the 24th delivery as both the opening bowlers Edward Rainsford and Tawanda Mupariwa made life hard for Bangladesh.
Rainsford, who conceded only five runs in his first six overs, earned the tourists their first breakthrough, trapping opener Zunaed Siddique plumb in front for three.
Was worse to come for Bangladesh as they lost the other opener, Tamim Iqbal, and skipper Mohammad Ashraful for 31 runs on board. Tamim scored 13 off 55 balls while Ashraful took 18 balls for his four.
Roqibul Hassan struck Price for two boundaries in successive balls before giving the spinner his first wicket with the next ball, a return catch for 28, the top score for Bangladesh.
After Price ran through the middle order Dabengwa cleaned up the tail.