Mike Modano also scored, while Marty Turco made 25 saves in the Stars' seventh win in their last eight trips to Silicon Valley. They've thoroughly dominated the Pacific Division champion Sharks here for two years, leading Modano to jokingly rename his team after their sixth straight win in January.
Jonathan Cheechoo scored the tying goal for San Jose with 3:02 left in regulation, but the Sharks' defensive sloth allowed Mattias Norstrom to whip the puck across the ice to Morrow early in OT. The Dallas captain scored easily against an off-balance Evgeni Nabokov, who finished with 16 saves.
Game 2 is Sunday night at the Shark Tank, with Game 3 in Dallas on Tuesday.
Morrow has five goals - already two more than his previous career postseason high
- in seven playoff games.
After eliminating defending Stanley Cup champion Anaheim in the first round, the fifth-seeded Stars rallied from a two-shot first period and an early deficit to take away home-ice advantage from the division rival Sharks, who needed seven games to put away Calgary in the opening round.
Milan Michalek scored an early goal for the second-seeded Sharks, who blew Game 1 for the second straight series. San Jose again endured special-teams struggles while taking a handful of careless penalties and struggling to generate offense against the Dallas trap.
The Sharks reached the second round for their NHL-best fourth straight season with a seven-game victory over Calgary. Meanwhile, Dallas ended a streak of three straight first-round defeats by eliminating the Ducks in impressive fashion.
Both teams came out cautiously for their ninth meeting of the season. The Stars didn't get their first shot of the game until late in their second power play with 3:36 left in the first period.
Michalek, who went scoreless in the Sharks' first seven games against Calgary, got credit for their first goal when his shot trickled underneath Turco midway through the second period. Modano, who scored four goals in four trips to San Jose during the regular season, tied it with a power-play goal 76 seconds later, putting a one-timer through traffic.
Morrow then scored on a rebound of Ribeiro's shot for his team-leading fourth goal of the postseason. The Sharks couldn't break through until Cheechoo, the goal-scoring specialist who also tied Game 4 against Calgary with a remarkable late score, banged home Torrey Mitchell's rebound of Matt Carle's shot for his fourth goal of the postseason.
