The Rams had gone into the half-time interval holding a 20-18 advantage thanks to tries from Francis Maloney, Alex Bretherton and Dean Lawford, along four goals from Malony.
The Centurions went toe to toe with their opponents, taking a second-half lead of 34-24, but a try from Andrew Bostock kept the home side in contention, while Josh Weeden and Kevin Crouthers touched down in the last ten minutes.
Maloney took his kicking record to six goals and it was his final kick of the game that proved the difference between the two sides and put the Rams into the fourth round draw.
In the round’s other all-National League clash, Whitehaven had a relatively easy time of beating Doncaster Lakers at the Recreation Ground, tries from Richard Fletcher (2), Carl Sice and Scott McAvoy helping Haven to a 24-10 win.
Gateshead Thunder beat Limoux 38-22 in a competitive game, Blackpool Panthers suffered another defeat at the hands of French opposition, 42-18 to Pia, and Swinton Lions put Lokomotive Moscow to the sword, 60-20.
Drighlington could also be pleased at their showing, given their relatively lowly Pennine League status, losing 52-10 to National league Two title chasing Featherstone Rovers, having trailed just 28-10 at the break.
Oulton Raiders went down narrowly 18-10 at the hands of Workington Town, East Hull suffered a 26-10 defeat to Oldham, Thornhill Trojans trailed Keighley Cougars 10-6 at the break before going down 34-6, West Bowling lost by just 24 points to eight against London Skolars, and Saddleworth Raiders trailed just 18-6 to Rochdale Hornets before finally losing 48-6.
Meanwhile, the much-anticipated derby between Castleford Tigers and Castleford Lock Lane ended in an 88-10 win for the National League One side, after their National Conference League Two opponents had trailed 36-4 at the interval.
Celtic Crusaders scored a 50-10 win over Eastmoor Dragons, Batley Bulldogs beat Widnes St Maries 60-6, West Hull lost 70-18 to Barrow Raiders, Widnes Vikings lost 78-0 at Widnes Vikings, and Halifax beat Eccles and Salford 86-12.