Carrick Rangers moved up to ninth in the Carling Premiership after a lacklustre stalemate at Seaview.
The hosts created the better chances in north Belfast with Paul Heatley firing wide after six minutes.
Heatley came close again when his shot was kept out by the feet of Swifts keeper Niall Morgan.
The best opening of the game fell to Rangers midfielder Anto Lagan on 65 minutes but his half-volley was well saved by Morgan.
Lagan was denied at the end of a good move involving Tony Anderson and Paul Owens.
Dungannon have moved level on points with Glenavon but remain 11th on goal difference.