The Cardiff region threw everything at the visitors and perhaps deserved more than a losing bonus point.
If Munster do the double over the Blues in December, the Tigers would have to turn the record books upside down and win at Thomond Park to challenge for top spot in the pool.
Blues skipper Martyn Williams admitted his side faced "a massive task" against Munster in December.
"We always knew it was going to be tough to get out of this pool, but I think we have shown that we can compete with anyone. We just have to pick ourselves up, dust ourselves off and wait for Munster in the next round. That will obviously be a massive task.
"I just think Leicester were a little bit more clinical than us today. We had our chances, but the commitment was there our guys and we weren't too far away."
The Blues showed grit in front of a bumper 26,000-strong crowd at the Millennium Stadium yesterday, but their lineout gradually got worse and their scrum spent much of the second half going in the wrong direction - not helped by a 50th-minute red card to prop Gary Powell for a head-butt.
Blues fly-half Nicky Robinson - unwanted by Wales coach Gareth Jenkins this autumn - opened the scoring with a clever chip over Tigers skipper Martin Corry, but it took the Tigers barely two minutes to hit back.
Centre Ollie Smith finished off a move started by a lack of communication at the back of a retreating Blues scrum, and Andy Goode added the extras before producing a moment of genius. A Blues midfield move broke down and Robinson was forced to scramble clear. The ball fell to Goode, who glanced at the other 29 players scattered across the field and delivered a perfect cross-field kick to give winger Tom Varndell the easiest of tries.
Ironically, Goode's poor clearance kick moments later gave Tom Shanklin the chance to take on the defence and flick the ball to Wales winger Chris Czekaj to score in the corner.
Leicester took a 15-14 lead into halftime, but they lost Smith in the process, the talented centre being helped off with a serious-looking knee injury.
Goode edged his side further ahead with his second penalty goal, before Powell aimed an ineffectual headbutt at George Chuter. Referee Alan Lewis had no option but to send him off, but made the game 14-a-side moments later when Varndell was yellow-carded for a blatant professional foul.
Fullback Ben Blair made it a one-point game with a pressure-packed penalty goal, but the Blues' handling let them down at crucial times, and a Sam Vesty dropped goal gave the Tigers some much-needed breathing space in the dying stages.