Cogliano, who signed a three-year contract with Anaheim on Jan. 12, 2018, scored in his final game with the Ducks, a 4-3 overtime loss to the Winnipeg Jets at Bell MTS Place on Sunday.
He is expected to be in the lineup when Dallas plays the Tampa Bay Lightning on Tuesday (8:30 p.m. ET; FS-SW, SUN, NHL.TV), according to the Stars website.
"I'm excited. He's played a lot of big games," Stars coach Jim Montgomery told NHL.com. "He's a veteran player who has had a lot of experience, a lot of success in the League that is going to add pace to our game, someone that can stretch the ice for us and also someone who is going to be very reliable in all three zones without the puck."
Ducks general manager Bob Murray, whose team has been outscored 40-19 in the losing streak, said the reason for the trade was simple.
"We were trying to get younger, obviously, but we're also looking for people with a little bit more creativity because we're not scoring enough goals," Murray said.
Shore, a 24-year-old forward, was Dallas' second-round pick (No. 64) in the 2012 NHL Draft. He has 17 points (five goals, 12 assists) in 42 games this season and 82 points (29 goals, 53 assists) in 209 NHL games. He signed a two-year contract with the Stars on July 13, 2018.
"It's tough leaving the organization that drafted you, but I'm looking forward to the new opportunity," Shore said.
"Everyone talks about how, once you get to the pros, it is a business, and there's been a lot of firsts today. First time feeling certain emotions and certain things, but then you kind of think it won't happen to you. But through the course of a career, it's definitely the exception to be with one team your whole career and not the rule. So you just embrace it and be a good pro and just remember that, even though it stings at the start, you're still playing the game you love for a living, and we're going to a great city and a great team, so we're fortunate there too."
The Stars (23-19-4, 50 points) are third in the Central Division, eight points behind the second-place Nashville Predators.