In one of the few games to beat the snow, York were in control 20-8 at half-time.
Judah Mazive crossed twice, while there were other York tries for Ben Cockayne and Tim Spears. Connor Robinson kicked five goals.
Swinton responded with tries from George Tyson and Rhodri Lloyd, Chris Hankinson kicking two goals.
Tries from Matty Fozard, Iliess Macani and Ryan Millar could not prevent Sheffield Eagles from sliding out of the Cup, losing 28-16 at Barrow Raiders.
Oscar Thomas and Simon Brown chipped in with goals, but it was not enough for the Eagles to progress.
Barrow’s tries came from Ryan Fieldhouse, Joe Bullock, Jarrad Stack, Gene Orsmby and Andy Litherland.
Dewsbury Rams suffered a 25-18 Cup exit at Whitehaven.
Martyn Reilly, Jack Teanby and James Glover all crossed for Rams touchdowns, the latter also kicking three goals.
But a brace for Jessie Joe Parker, plus tries fro Carl Forster and Jason Mossop, four Jordan Burns goals, and a Stuart Howarth drop goal, saw the hosts progress.
Amateur side Normanton Knights saw their brave cup run end, losing 20-8 to Rochdale on Saturday.
Full-back Connor Wilson cross for a try in each half, but it wasn’t enough for the Wakefield club against Championship opposition.