The Aviva Premiership champions were beaten 22-16 by the Irish outfit at Ravenhill in their opening match of Pool 5 on Friday night.
Tigers picked up a consolation bonus point thanks to a late Owen Williams penalty kick, which pleased Burke.
"When our backs were against the wall, we showed great character and resolve and that is the mark of a champion team," Burke told reporters. "Defensively, we were heroic at times. We were not happy to just fall over and concede tries when we were up against it.
"European rugby is a step up and to [go to Ulster] and play behind a passionate crowd was a big test. When Jordan Crane was in the sin-bin, we never let our heads drop and we kept going right to the end.
"It has been well documented how many injuries we had going into the game. We had only 10 fit backs for this fixture. Against that backdrop, it was an outstanding and heroic effort from the guys who were out there, and that will stand us in good stead for the rest of the competition."
Leicester host Treviso in the competition on Friday.