The Premiership side scraped into the quarter-finals as Pool Three winners despite this defeat.
A losing bonus point was enough but Tigers full-back Murphy does not expect to have home advantage after today's other final pool games.
Tigers have not won any of their last seven away games and man-of-the- match Murphy said: "It's a bit of a strange feeling to have qualified despite losing.
"We came to win and we're disappointed - but at least we're in the quarter-finals. Our away form this season is something we have to sort out - especially as we expect to be away in the quarter-finals.
"I think the others team will overtake us so it's going to be tough."
Ospreys must also await the outcome of today's games to see if they are one of the top two runners-up.
Five James Hook penalties gave them the win with Tigers replying with three from Derrick Hougaard.
Ospreys smashed into Tigers at every opportunity and this Leicester side is a shadow of Martin Johnson's double Heineken Cup winners.
They are weak at half-back and lack physicality in the backs, apart from Seru Rabeni, and the Ospreys were not scared to rough them up a bit.
Leicester did not do themselves any favours when Julien Dupuy was pole-axed by team mate Rabeni as the pair chased a Gavin Henson chip and chase as the Ospreys pressured.
Hook booted a penalty after Andrew Bishop had cut Rabeni in half and he added a second on nine minutes. But the hosts could not make any inroads into the Leicester defence and Hougaard narrowed the gap.
Finally on the half hour the Ospreys made it into the Leicester 22 and after a half-break by Shane Williams were poised to go for the line.
But Toby Flood's cynical foul denied them a score and Hook's penalty sailed past the upright. Flood, potentially England's Six Nations fly-half, had barely featured and was binned.
Ospreys piled on the pressure near the break. Henson went close and then they wasted an overlap on the right. Ospreys also had the best of the second half but Hook and Hougaard were the only ones to get more points on the scoreboard.
SCORERS - Ospreys: Pens: Hook (5) Leicester: Pens: Hougaard (3)
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