The 26-year-old, who scored 14 times in 72 league starts for the Canaries, signed on the line for last season's Championship runners-up after several days of talks at the JJB Stadium.
Francis, a former Wimbledon player, joined his new teammates for training this morning as the details of the deal were announced. Norwich are guaranteed £1.5million for the midfielder, with £500,000 payable after a certain number of appearances and another £250,000 if Francis helps his new club to stay in the Premiership.
Francis' position at the club had become untenable after he was withdrawn from a friendly with Colchester United. At the time, it was expected that he would join Portsmouth but Wigan, who have struggled to bring in new recruits since winning promotion, stepped in.
Although the manner of his departure has left a sour taste in the mouth of many fans, Damien was an important player in his two seasons with Norwich. He scored his fair share of goals from midfield and was sorely missed last season when he suffered a fractured cheekbone in a training ground clash with Leon McKenzie.
The vacant berth in the Norwich line up is likely to be taken by Jason Jarrett, who moved in the opposite direction at the start of July. Jarrett had publicly urged his colleague to stay at Carrow Road, but will now have to step into his shoes as City attempt to win promotion back to the Premiership at the first attempt.