Dons skipper Dean Lewington broke the deadlock with a 55th-minute header and striker Jermaine Easter grabbed a double before substitute Josh Scott pulled a goal back for the Daggers at the death.
Robinson said: "We take all competitions very seriously, whether it is a five-a-side game in training, a league or cup game. The way we played was fantastic.
"It was a typical League Two game for us from a couple of years ago. We struggled to break them down in the first half but at half-time we asked them to up the tempo and be more aggressive with their passing and that's what they did.
"We kept knocking on the door and the lads were rewarded when Lewi gave us the lead - he took his chance well. When we signed Jermaine we knew he was a goalscorer. He scores all different types of goals - one was from outside the box and the other was a tap-in, which is just what needed.
"The manager wasn't happy with certain parts of our game but he was happy with the lads' patience."