Robinson saw his side take an early lead before succumbing to goals from Kieran Trippier and Andre Amougou in the second, but he said after the game: "I am very, very proud of the young kids that have come here and played football in a way that I still think is unique in this country.
"We played with freedom and expression, our goal came from a one-two in their penalty box and it was a wonderful finish from 20-year-old Powelly (Daniel Powell).
"It's about playing it the right way. People will argue and say it's about results and I respect that but with the right performances nine times out of ten the results will follow. We had some experienced players but there was an awful lot of youth as well.
"I want to build a philosophy and an infrastructure that stands this club in a good stead for many years to come. Today shows from the youth team players to the first team there is a philosophy and a grounding that will stand the club good stead in future years.
"I finished the game with no Jay O'Shea, Stephen Gleeson, Darren Potter, Angelo Balanta, Charlie MacDonald, Gary McKenzie, Mark Beevers, Lewis Guy, Dean Bowditch or Mathias Kouo-Doumbe on the pitch. They were missing through injury, suspension or were cup tied.
"That's a lot of quality at this level in League One and a lot of quality to be missing from the team and to still be able to put in a performance like that, I'm so proud of them."
Robinson was very complimentary about the Clarets' winner from Andre Amougou. He said: "I started clapping when that overhead kick went in the bottom corner. I'm not too sure if it took a deflection because I've still not seen it yet but his thought process to change his body position to get himself in that position to put it into the bottom corner shows the quality of the player."
Eddie Howe saw his team win through and said: "We are pleased that we got the job done. We conceded early again and the confidence drained from the players in a scrappy first half. But the confidence of a goal gave the lads a lift and in the last half hour we showed what we are capable of.
"It's our first win at home in 90 minutes and that will give us a boost. Later on it looked like extra time again, but a special goal won it. It's not something I've seen Andre do too often, but it was a fantastic finish at a great time and I thought we deserved it on our second half performance.
"It was a great goal, in fact we scored two really good goals and we could have had more. Our second half performance was a lot better and we moved the ball quicker, got the ball wide and got crosses into the box.
"Off the ball we were better too, and the really pleasing thing from my point of view is that heads didn't drop at 1-0 down. We showed spirit and hopefully, now, we can take that second half display into Saturday."