Guinness Premiership regulations stipulate clubs can only field one non-EU qualified player at any time.
But Newcastle can now field both Burke, their captain, and New Zealand prop Joe McDonnell in the same team.
Burke said: "The opportunity arose for me to help out the Falcons, who have been so good to me during my time over here, and with mum being born in England we decided to explore the options further.
"Under English rugby rules only one foreign player can be on the field for a team at any one time, so having gone through the process now it gives the Falcons more options in terms of selecting their side each week."
Burke moved to Newcastle two-and-a-half years ago from the NSW Waratahs and lives in Gosforth with his wife Kate and two daughters.
Falcons director of rugby John Fletcher said: "To go through such a lengthy and complicated bureaucratic process speaks volumes of Matt and the type of bloke he is.
"Him becoming a British citizen helps the club massively because it allows us to field both him and Joe McDonnell, two of our strongest players, at the same time. "
Burke was presented with his new citizenship by the Lord Mayor of Newcastle, Councillor Diane Packham, at a ceremony in the city's Civic Centre.