Tuilagi, the 25-year-old brother of England centre Manu, has long been linked to a switch to Rodney Parade.
However the Dragons have now been given the green light to sign the former Leicester, Leeds and St Helens centre.
Director of rugby Rob Beale said: "This has been a very long process for the Dragons and for Andy.
"I know there has been media speculation over the past couple of months, but we had to ensure that this deal was done properly and that Andy had the necessary documentation in place before the signing was made.
"Having said that, Andy's reputation speaks for itself and I am confident that the wait will be well worth it."
Having finally got his man, Dragons head coach Darren Edwards is looking forward to seeing the Samoa international in action.
"Andy is exactly what we want in our back line," said Darren Edwards.
"We were keen to look at the strengths we have within our squad and make careful signings based on players who we feel will add to our game. Having worked with Andy at Leeds, I'm very aware of his qualities as a player and am confident that he will be successful at Rodney Parade.
"We have a great deal of skill and pace in our backs and Andy will bring a physicality and a line-breaking ability that will really complement our current squad."
Tuilagi made 42 appearances over three seasons for Sale, despite a layoff with a broken arm.
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