Leinster were big favourites to beat the French side, who eventually prevailed in a thrilling final in Marseille.
And Saints will need similar underdog spirit on Saturday as they attempt to win in the Gallagher Premiership play-off semi-final at Welford Road.
Leicester won every home league game during the regular season and topped the Premiership table from start to finish.
That means travelling to face the Tigers will be a daunting task for Saints.
But Boyd said: "You don’t get a choice (about being named as underdogs) but I think you look at the sport that’s been about the place, even rugby, in the last couple of weeks.
“There weren’t very many people that fancied La Rochelle, but if you’re in you’ve got a chance.
“Our worst two performances this year have both been against Leicester, home and away they were shockers.
“We’ll have to play a whole lot better against them to challenge them.”
Saints also know they will have to play better than they did against Newcastle Falcons last Saturday.
Because even though Boyd's side scored 10 tries in a huge 65-26 win, there was a period after half-time when they were under huge pressure, with indiscipline costing them dear.
“That was clearly a long way from the script," Boyd said.
“The danger of a game like that, where you know that trying to win the game is so important to get that fourth spot, you invariably push the game too hard to try things.
“We knew that if we went forward and created platforms to work off, we’d probably get enough platforms to get a few tries, but we tried to force things too much.
“There’s probably not much out of that you take into Leicester. That’s going to be a completely different game.
“To be fair, two months ago I think we were sitting eighth or ninth and a long way away from looking fourth.
“We talked about the fact we were probably playing knock-out football and we’ve managed to just sneak into that top four.”
Tommy Freeman once again caught the eye against Newcastle, scoring a hat-trick.
There has been talk that the talented young back, who has been utilised on the wing and at full-back by Saints, could become an international 13 for England.
And Boyd said: “I actually think he can play in a number of positions.
“But at the end of the day, I think he’s got a very good skill set for full-back, he’s got a very good skill set for a winger and he could easily play at 13 as well.
“He’s big, strong, fast and he’s a pretty good footballer, so I don’t think it matters where he plays.
"He’s a good player and he’s not the complete article, either."