Alonso, 23, rejected a contract offer from former club Bolton Wanderers in favour of putting pen to paper on a three-year deal with Fiorentina in the summer.
The former Real Madrid defender has totaled nine appearances, including six starts, in all competitions for the Viola this season.
“I can confirm Marcos is going to be with us for the rest of the season,” said Poyet.
“I was looking for a player in that position on the left side and I hope he will help us and everybody will like him.
“I am delighted that everything is agreed.”
Alonso will compete with the likes of Andrea Dossena and Jack Colback for the starting left-back job at the Stadium of Light.
Bottom-of-the-table Sunderland, who have won four points from their last two English Premier League fixtures, are backed by the bookmakers at 5/6 to be relegated to the Championship this season.
Rosler wants to keep Powell on loan at Wigan
Wigan Athletic boss Uwe Rosler would be absolutely delighted to keep Manchester United forward Nick Powell on loan at the DW Stadium until the end of the season, according to Sky Sports.
Powell, 19, has been on scintillating form for the Latics, in both the Championship and the UEFA Europa League, this term, having tallied eight goals and one assist in 24 first-team appearances.
The highly-rated England youth international has caught the eye of United manager David Moyes in recent weeks, so much so that the Scot is contemplating whether or not to bring him back to Old Trafford in January.
Moyes is scheduled to speak with Rosler regarding the future of Powell sometime this week, with a final decision expected to made before United’s clash against Tottenham Hotspur at the weekend.
“I’ve spoken to David and, so far, there’s been no indication that anything has changed,” said Rosler following the Latics’ goalless draw against Burnley on Sunday.
“On Monday and Tuesday we’ll have another talk and I’ll know for certain.
“Nick is a player of real quality and I would be delighted if he is able to stay with us for the rest of the season.”
Powell and William Keane, who is also on loan at Wigan from Manchester United, are 15/8 to score at anytime against Derby County at the iPro Stadium on New Year’s Day.