Ichiro Suzuki contributed his 197th and 198th hits of the season, leaving him two short of a Major League record ninth consecutive 200-hit season.
The series opener, which was rained out on Friday, started 17 minutes late because of lousy weather and was reasonably playable for the first seven innings.
Heavy rains returned, but the game was not interrupted until the bottom of the ninth inning.
With a runner on first and nobody out, Nelson Cruz hit a grounder to shortstop Jack Wilson that had the appearance of becoming a double play. But Wilson lost his footing and then tried to shovel the ball to second baseman Jose Lopez for an out.
The connection never was made and Marlon Byrd, aboard on a leadoff single, nearly slid past the bag because of the slippery conditions.
Wilson and Mariners manager Don Wakamatsu explained to the umpiring crew that the conditions were unplayable. After a brief huddle, the umps agreed and cleared the field.
More than one hour later, it was still raining hard and crew chief Gary Cederstrom decided enough was enough and pulled the plug.
"It was a great victory under those conditions," Wakamatsu said. "It was a routine ground ball that Jack slipped because of the field conditions. It got to the point where I thought somebody was going to get hurt."
"It wasn't too bad for a while, but by the fifth inning it was like pitching with a misting machine over your head," Morrow said.
The right-hander (1-4) finally got his first win of the season -- thanks to plenty of offensive support.
Ken Griffey Jr. had three hits and drove in two runs. The two Jacks in the lineup, Wilson and Hannahan, each had two hits. And so did third baseman Adrian Beltre.
Morrow was supposed to make his return a day earlier, but the extra day's wait was worth it.
"I wasn't as sharp as I would have liked to be," he said. "But it has been kind of a crazy week. I only got to throw one time between my last start down in Tacoma and here and then the rain delays and stuff.
"I was getting kind of anxious to get out there."
The Mariners used a first-pitch-of-the-game double by Ichiro and two-out single up the middle by Griffey to take a 1-0 lead in the first inning.
Ichiro drove himself in the next time he batted against Rangers starter Kevin Millwood, hitting a solo home run into the right-field seats in the third inning. It was his ninth circuit clout of the season.
Junior added another RBI in the same inning, driving in Jose Lopez from first base with a double into the left-center field gap and Hannahan drove in two key runs in the seventh inning, turning a two-run lead into a four-run gap.
Morrow weathered the storm and departed after 84 pitches. He walked two and struck out one.
"I wanted to show that I had good command of my offspeed [pitches]," he said. "I was doing a real good job of that down in Tacoma and that is something I wanted to continue up here. But like I said, I wasn't real sharp, but I had to make big pitches when I had to."
The biggest came in the in the fourth inning.
Morrow walked Cruz leading off and then induced left-handed-hitting David Murphy to ground into a 3-6-3 double play. He then struck out Ivan Rodriguez to end the inning and protect a 5-2 lead.
"That double play won the game for him," Wakamatsu said. "He fell behind six of the first 12 batters he faced and most of them were fastballs. It seemed a little bit like he was going back to where he was before, trying to force something that's not there.
"I talked to both he and Rob [Johnson, catcher] about it. I just didn't think he attacked hitters and that's something that will very important for him. I saw him a couple of pitches that were 95, so he really turned it loose. But he did use his offspeed pitches more, and that's good."
Morrow said the crucial double-play pitch was a changeup.
Wakamatsu was hoping to get closer David Aardsma in the game, but with a five-run lead in the ninth, and the playing conditions deteriorating by the minute, he decided to give the final three outs to Randy Messenger and save his best game-ender for another day.
"With two games tomorrow and the possibility of saving two, that's the way we went," he said.