Franklin Gutierrez sparked the winning rally when he led off with an infield single. He advanced on Michael Saunders' sacrifice and came home when Morales drove the next pitch from Nate Jones (0-1) down the first-base line.
Montero then singled in pinch-runner Robert Andino as Seattle earned its fourth win in its last 20 games in Chicago dating to the 2008 season. The Mariners have only 11 wins in their last 44 games against the White Sox overall.
Gutierrez also homered and drove in three runs for Seattle, which blew a 6-1 lead on a chilly night at U.S. Cellular Field. Kameron Loe (1-0) pitched a perfect ninth for the win and Tom Wilhelmsen barely held on for his second save.
Dewayne Wise hit a two-out RBI single for the White Sox in the 10th, leaving runners on the corners. Alexei Ramirez then walked to load the bases, but Tyler Flowers struck out to end the game.
Alejandro De Aza and Alex Rios homered for Chicago, which went 8-1 against Seattle last season. Paul Konerko had an RBI single in the fourth for his 2,137th hit with the White Sox, passing Frank Thomas for third on the franchise list.
The White Sox were hurt by pitcher Jose Quintana's costly error that paved the way for Seattle's five-run fifth inning. With none out and a runner on second, Dustin Ackley hit a grounder to first and Quintana hustled over to cover the bag. The throw from Konerko was a little behind him, but the left-hander tried to grab it with his bare hand and it squirted away.
The Mariners then strung together five consecutive hits to chase Quintana and open a 6-1 lead. Gutierrez had a two-run double and scored on Saunders' triple. Morales added an RBI single, stopping a 1-for-14 slide.
But the White Sox responded with four runs in the bottom half and another in the seventh. De Aza hit a two-run shot and Rios had a solo drive in the fifth, and Rios' grounder brought home Gordon Beckham with the tying run two innings later.
Brendan Ryan also had two hits and drove in a run for Seattle, which finished with 12 hits overall.
Gutierrez got the Mariners off to a nice start when he drove Quintana's fourth pitch just over the wall in the right-field corner. It was Gutierrez's second leadoff homer of the season and No. 3 for his career.
NOTES: The game-time temperature was 34 degrees. ... White Sox LHP John Danks will pitch in an extended spring training game again on Tuesday. Danks, working his way back from August shoulder surgery, threw five innings of one-run ball on Thursday in Arizona. "Listen, he's ahead of schedule but he needs more time," pitching coach Don Cooper said. "It's a shoulder injury and nobody knows the answer to these things and we're all going to hear on an every five- or six-day basis." ... Mariners ace Felix Hernandez will make his second start on Saturday. The right-hander gave up three hits over 7 2-3 innings in a 2-0 win at Oakland on opening day. He will be opposed by RHP Dylan Axelrod, who will be making his season debut. ... With the short turnaround between games, Konerko likely will get a day off on Saturday. Chicago manager Robin Ventura also said C Hector Gimenez probably will play. ... Mariners manager Eric Wedge said C Kelly Shoppach likely will be in the lineup on Saturday.