The Stars hold the NHL record with 50 consecutive wins when leading after two periods, setting the mark with a 5-2 victory at Washington on March 12.
But in its last five games, Dallas has trailed after 40 minutes and still has earned a point in each, going 2
The Stars trailed for 58:46 until a backhander by Gilchrist tied it 3-3. Working along the right boards, Gilchrist's shot went off the skate of Coyotes defenseman Paul Mara for his second goal since joining Dallas on Feb. 13.
Dallas took its only lead when a one-timer by Morrow trickled past goaltender Sean Burke.
The Coyotes built a 3-1 lead on Daymond Langkow's 21st goal 3:04 into the third period. Langkow outhustled defenseman Sergei Zubov to the net and fired the puck just past the outstretched arms of goaltender Ed Belfour.
Dallas is tied with Phoenix for seventh place in the Western Conference. Both teams have 80 points, only three behind Los Angeles and Chicago for the fourth seed.