Morrow scored the go-ahead goal at 1:13 of the third period, Andrew Raycroft made 18 saves and the Stars beat the Senators 2-1 on Wednesday night.
Trevor Daley also scored to help Dallas snap a four-game road losing streak.
"I think that was a great road game, almost a perfect road game," Raycroft said. "I was able to get in front of a couple and not give anything bad up and get a win."
Dallas is 11-8-1 overall.
"I was pleased with the way we played and the guys are pretty happy about it, too, so hopefully it will bode well for us on the road in the future," Stars coach Marc Crawford said.
"We're just not putting enough pucks on net," Ottawa coach Cory Clouston said.
The Senators have lost four of their last five.
"It was not a very exciting game at all," Senators forward Mike Fisher said. "[Leclaire] kept us in for a lot of the game and made some big saves and we didn't generate a lot as far as quality chances and when we did Raycroft was there. We just didn't seem to have any spark or jump that we needed."
Daley opened the scoring at 6:14 of the second period, beating Leclaire high on his glove side.
The Senators tied it with 3:31 left in the second on an ugly goal. Shannon shot from just inside the blue line and the puck ended up bouncing behind Raycroft.
Dallas had a great chance to end the period with the lead, but Leclaire stopped Morrow on a breakaway with 35 seconds remaining.
In the third, Morrow scored his ninth goal of the season after some strong work by Mike Ribeiro to maintain possession deep in the Senators' end. Ribeiro spotted Morrow at the top of the slot, and the Dallas captain quickly beat Leclaire with a wrist shot.
Morrow scored for only the second time in eight games.
"For the people who have seen us a lot this year, Brenden has been marvelous and it was great to see him score tonight," Crawford said.