The Bluebirds needed a late strike from Peter Whittingham to settle their nerves as Dagenham battled hard.
Gavin Rae and Jay Bothroyd from the spot had put the home side ahead but when Josh Scott pulled one back nine minutes from time for the League Two side it made for a nervous closing few minutes before Whittingham restored the two-goal cushion.
"Every time we moved it around we caused them problems but when we slowed it down they got into us and full credit to them," said Dave Jones.
"We have had Dagenham watched several times and they are a much improved side and they came here and put up a good show."
Rae had opened the scoring in the 20th minute when the impressive Whittingham slipped a pass inside to him and he fired high into the net from fully 20 yards.
Five minutes later the Bluebirds doubled their lead from the spot. Bothroyd was brought down by Will Antwi but his poorly taken penalty was easily saved by the Daggers Welsh keeper Tony Roberts.
Referee Andy Penn signalled a retake for encroachment by Dagenham and then Bothroyd and Michael Chopra had a rare old tussle for the ball as Chopra also wanted to take the kick.
Bothroyd held on and then drove the spot-kick home to extend the Bluebirds' lead.
Despite the goal, Jones admitted Chopra had a case for taking the penalty the second time around.
"I have no problem with Michael Chopra wanting to take the penalty from Bothroyd after the referee ordered a retake," he said.
"Bothroyd's first attempt was very poor and I would expect every other player to want to take it, not just Chopra."
The visitors penned City back in the early stages of the second half but were unable to carve out any clear openings and Peter Gain had their best chance but fired wide when well placed.
The visitors pulled their goal back when Scott met a superb cross from Antwi to turn the ball into the net beyond Peter Enckelman for the first opposition goal at the new Cardiff City Stadium.
The Bluebirds settled the match when Whittingham exchanged passes with Stephen McPhail before beating Roberts from close range.