Gnonto gave Leeds a flying start with a blistering volley in the opening minute at Elland Road.
The Italian striker bagged his second after Rodrigo netted his 11th goal of the season as Leeds raced into a 3-0 lead before the interval.
Bamford came off the bench to score twice in the second half before Callum Robinson grabbed two late goals for Cardiff.
Leeds are into the FA Cup fourth round for the first time in six years and now face a trip to Accrington or Boreham Wood.
A stoppage-time equaliser from Leeds teenager Sonny Perkins had salvaged a replay at Cardiff 10 days ago, but the managerless Championship strugglers were never in the contest this time.
After sacking Mark Hudson on Saturday, it was Dean Whitehead who was in charge of Cardiff on an interim basis and his reign couldn't have got off to a worse start.
Rodrigo clipped an inch-perfect ball into the penalty area and Gnonto scissor-kicked his volley into the roof of the net after just 28 seconds.
Jesse Marsch's men struck again when Rodrigo cut inside to curl home in the 34th minute and Gnonto drove in his second moments later following an incisive counter-attack.
Two more half-time substitutes combined to put Leeds four up in the 71st minute as Bamford ran on to Luke Ayling's pass to deliver a cool finish.
Bamford notched his second to make it five minutes later, nonchalantly finishing with the outside of his left boot.
Robinson gave the scoreline a more flattering look for out-classed Cardiff when he heading in Sheyi Ojo's cross and converted a stoppage-time penalty after Marc Roca's handball.