"I drove up through the night to be here so it's been a manic 24 hours but it's all been worth it to come here and play for this great club," Shelvey told United's website on Tuesday.
The transfer fee was undisclosed but media reports said Newcastle had agreed to pay 12 million pounds for the 23-year-old.
He is Newcastle's second signing of the January transfer window after Senegal midfielder Henri Saivet arrived on Monday.
Shelvey, who has six England caps, joined Swansea from Liverpool in 2013 for a reported fee of five million pounds.
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He made 79 league appearances and scored 10 goals for the Welsh club who are fourth from bottom, two points clear of the relegation zone.
Despite Swansea's difficult season, Shelvey often found himself on the fringes of the side and made only 14 league starts this term.
He has started only three league games since October and was left out of the squad for the recent matches against Leicester City, Manchester City and Manchester United.
Shelvey, named in England manager Roy Hodgson's squad for the Euro 2016 qualifiers against San Marino and Switzerland in September, is swapping one relegation battle for another with Newcastle third from bottom.
"He has great experience for a young English central midfielder and is a current England international so I'm certain he will be a great addition to the squad," said Newcastle manager Steve McClaren.
"He is a player we have admired for some time and we're very pleased we've now been able to bring him here."
Newcastle host Manchester United in the league later on Tuesday.
(Reporting by Simon Jennings in Bengaluru and Toby Davis in London, editing by Ed Osmond)
