Pool Four leaders Ulster stand two points above Leicester with just two group games left, while French heavyweights Clermont Auvergne also have realistic last-eight aspirations, lying just a point behind Murphy's men in third.
If Ulster win at Ravenhill (kick-off 8pm) - and their previous European scalps there include Leicester, Clermont, Toulouse, Biarritz and Stade Francais - by more than seven points, the Tigers are effectively looking at elimination at the pool stage for only a fifth time in 15 European campaigns.
Although Leicester's final fixture is at home to Italian strugglers Aironi next weekend, the damage could easily have been done by then, especially as they will travel without injured England trio Toby Flood, Manu Tuilagi and Louis Deacon.
"Ravenhill is a fortress for them, they have beaten pretty much every top side at home," full-back Murphy said.
"Ulster are very successful at Ravenhill, especially on a Friday night. Traditionally it's wet and windy and the wind howls through the ground, making it difficult for opposition goalkickers who don't know the ground.
"It will be no different to any final or any game as big as that; we are going to have to put in a very good performance up front, be defensively sound to stop them scoring tries and kick our goals.
"It will be an arm wrestle, and hopefully we get a bit of luck."
Leicester boss Richard Cockerill has called props Marcos Ayerza and Martin Castrogiovanni into his starting XV, while Ulster are boosted by their former Tigers fly-half Ian Humphreys and back-row forward Pedrie Wannenburg both being declared fit following injuries.
Murphy added: "It has turned out to be a very tight and difficult group. I knew when I saw the draw that it was going to be very difficult to get out of the pool.
"Ulster are a very, very strong side at home and Clermont are also very difficult to beat. Clermont's bonus point (in Belfast) could be costly, and so could our lack of a bonus against Aironi.
"We have to go to Ulster and get a win. To have the group in our control, we have to go and get two wins in the final two games.
"Things have gone reasonably well for us recently, but the schedule has also taken its toll on us physically. A few guys will be out for the Ulster game, and while that is a bit of a nightmare, there is nothing you can do.
"They will be missed, but that is the way the cookie crumbles. We just have to get on with things and do as good a job as possible.
"The strength in depth seems to be there and we have some pretty good players who have come in and done a good job, so we have just got to keep grinding out wins."
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