Trocheck gave New York a 2-1 lead 59 seconds into the second period with a power-play goal.
"First (goal), on the power play, it was just a great seam pass from Mika," Trocheck said. "[Panarin] wasn't out there, so I just tried to find some open ice on the other side."
Roope Hintz tied the game 2-2 at 1:28 when he capitalized on a rebound at the front of the net.
Zibanejad gave the Rangers a 3-2 lead at 14:18, tapping in an Artemi Panarin pass in front of the net on the power play.
Robertson tied it 3-3 at 18:46 of the second when he won the puck off the face-off and beat Shesterkin five-hole from the slot.
Robertson then appeared to give Dallas a 4-3 lead 3:51 into the third, but Gallant challenged the play for offside, and the call on the ice was overturned.
"We didn't do enough to dictate the game for sure, and you want to dictate here at home," DeBoer said. "They should have to adjust to us, and I didn't think that was the case."
NOTES: Shesterkin (4-0-2) became the first Rangers goalie to start the season without a regulation loss in his first six games since Eddie Mio (5-0-0 with two ties in 1981-82). … New York was 2-for-5 on the power play. … The Stars lost at home for the first time this season (3-1-0). They had outscored opponents 11-2 in their first three home games. … Rangers forward Vitali Kravtsov sustained an upper-body injury at 12:27 in the second period after battling for the puck with Stars defenseman Jani Hakanpää in the corner. Gallant said he is day-to-day.