The new five-year contracts will keep the duo at the Amex Stadium until the summer of 2016. Their previous deals was set to expire in 2014.
Poyet told the club website: “The new contracts will allow us to make even more long-term plans, particularly with the new academy and training ground.
“I am delighted. The last two years here have been absolutely fantastic.”
He added: “Both Tano [Taricco] and I would like to thank the chairman and the directors – and also my staff and the players, who have been crucial in our progression as a club.”
Albion chairman Tony Bloom said: “This is great news for the club. Gus and Tano are key to all of our immediate and long-term plans.
“When you consider the position the club was in when they arrived, what they have achieved here in less than two years is outstanding.
“This gives the club even more long-term stability and further underlines their own commitment to the club.
“Gus and Tano have brought us success, and in doing so revolutionised the way the Albion play; giving us a brand of football the supporters love watching.
“I am certain they will continue to be successful and the club will continue to progress with them in charge of the team.”