A losing bonus point in a 15-9 defeat at the Liberty Stadium took Tigers to 21 points from six games and secured a place in the last eight for a seventh time in nine seasons.
“At the start of the tournament we set out to get into the quarter-finals and that bit is done,” said Tigers skipper Corry, who made his club record 69th European appearance in Swansea.
“Ospreys are an exceptional side, and we were very disappointed to lose the game. But we did enough to qualify.
“The attitude of the players was fantastic. We had to defend for our lives at times but not many teams keep them out and nil their try count.
“We are happy to be through to the next round.”
Tigers scored 23 tries in their four pool wins – a score equalled only by top seeds Cardiff Blues.
“It was a tough group, Ospreys are in good form, and Perpignan,” said Corry.
“And for us to get through as group winners we take huge amount of pride in that. Look at some of the teams that have fallen at the first hurdle. It is important that we are through in the first place in the group.”
Respective fly-halves Derick Hougaard and James Hook shared the points at the Liberty Stadium, and Corry identified the quality and spirit of the Tigers defensive work as a crucial part of their play.
“Probably the key part of the game was the way we defended before half-time when we were down to 14 men,” he said.
“The Ben Woods steal at that stage was vital.
“The way we defended and the attitude summed up absolutely everything.
“There is probably not a better attacking side in the competition than the Ospreys and to hold them out with 14 men was a fantastic effort.
“Then the defence in the last 10 minutes, when it was so close, was very important.
“They were attacking around our 10-metre line, but we had resilient defence. James Hook had slotted a couple of long kicks in the first half and we knew we could not give away any penalties there.
“The discipline of the defence and aggressive nature of the defence to not get much further than the 10-metre line was very heartening.”
Tigers now face Bath Rugby in the Heineken Cup quarter-finals, to be played on the weekend of April 10-12.