The Senegal international completed his medical on Tuesday before personal terms were agreed and a work permit was secured on Wednesday.
The 27-year-old has been capped seven times by his country and will be eyeing a starting berth for the Potters' trip to Bournemouth on Saturday.
"It's a very good team in England, so I'm very happy and proud," Ndiaye told Stoke's official website.
"I'm really, really happy. As I said, I wanted to join Stoke City. Today I'm here, I truly hope I will have a good time here and help the team do well."
Ndiaye has made 17 league appearances for Galatasaray this season, scoring one goal and adding three assists.