The 26-year-old, who predominately plays at centre-back, has been with the Ligue 1 club for the last four years, in which he played 118 times and scored four goals.
Ecuele Manga is Ole Gunnar Solskjaer's ninth signing of the summer.
Solskjaer said: “We’ve been waiting for this signing – Bruno is someone we’ve watched for a long time now.
“We faced stiff competition for his signing, but managed to get the deal done. We’re delighted; I said we would only add more before the window shut if something spectacular came up for us. It did and Bruno will be a terrific addition to our squad.
“Bruno will bring pace, international experience, energy and enthusiasm. He is a centre-back who plays the ball forward, looks up and can put us on the counter-attack. I’m delighted he’s agreed to join us.”
Ecuele Manga added: “I spoke at length to the manager who was a great player in this country. He has certain ambitions and I share those ambitions. The main one is to go back up to the Premier League, and I'm very excited about coming here.
“I’ll spend time away with my national team now, though I look forward to returning and starting with my new team-mates. I’m confident that things can go very well for me here. This is a good team with a good manager and a good board that is doing great work. I can’t wait to start."