Beauxis was named Amlin man of the match in the 29-25 victory and Cheika will now hope his star man produces a similarly devastating display in his final match in Stade colours at the Cardiff City Stadium on May 20.
"When he's on song, he's unstoppable," said Cheika, after Beauxis led Stade to a first European final in six years.
"That's with his kicking game and his running game. And his goal kicking was absolutely out of this world.
"He's just so natural. He's a great player and Toulouse are lucky they're getting him because he's class. As long as he doesn't do it against us next year, I'll be happy!"
Beauxis was impressive throughout at the Stade Charlety, pulling the strings and keeping Clermont pinned back with a clever kicking display, while his opposite number Brent Russell endured somewhat more mixed fortunes.
Russell was superb at times but frustrating at others as he claimed a stunning solo try but gifted Stade two scores of their own.
Stade led 19-10 at the break thanks to Beauxis' boot after both sides had crossed for a try apiece.
Beauxis and Clermont's Morgan Parra were both off target with their first penalty attempts of the day but it was the Stade No10 who gave his side the lead with 12 minutes played. Parra matched Beauxis is succeeding with his second attempt at goal, though, as Clermont hit back almost immediately to make it three points apiece approaching the quarter-of-an-hour mark.
The game's first outstanding moment arrived just a minute later as Clermont cut Stade to shreds with a scintillating counter attack.
Club captain Aurelien Rougerie bounced off two defenders to take play 50 metres upfield from the edge of the Clermont 22 before flankers Julien Bardy and Jamie Cudmore made hard yards in home territory.
Clermont initially switched play to the right before swinging the ball back to the left where quick hands allowed livewire full back Tusiata Pisi to dive over in the corner. Parra added a superb conversion to suddenly move Clermont seven points clear.
Russell's jinking midfield run then looked as though it might lead to an overlap for Clermont but the former Springbok Sevens star instead threw a loose pass to gift Stade an interception try.
Julien Arias was the grateful recipient, with the French international wing enjoying a free run to the line from just past halfway.
Beauxis followed the easiest of conversions with a hat-trick of penalties to hand Stade a 19-10 lead and he should have made it a 12-point ball game just after the break but, for once, he failed with a straightforward 35-metre penalty attempt two minutes into the second period.
Parra was not so profligate four minutes later as he cut the gap to six as Clermont began a mini-scoring spree.
The first three points of the half were quickly followed by seven more as Russell made amends for his first-half error in spectacular fashion as showed his slight of foot and searing speed with a superb individual score on 48 minutes.
The ex-Saracens and Stormers back made the most of being shown the inside shoulder just outside his own 22 before planting a beautifully-weighted kick behind the backtracking Stade defence. The 31-year-old then showed that age hasn't affected his sprinting skills as won the foot race to the loose ball to claim a hugely opportunistic try.
Parra added the extras from 15 metres to the left of the posts to give Clermont a one-point lead.
Only a heroic last-gasp tackle from Arias prevented Kini Murimurivalu from extending the gap to six, although the Stade try-scorer saw yellow for failing to roll away.
Parra failed to capitalize from the resulting penalty, however, and Stade soon hit back themselves when Beauxis slotted a 55th-minute penalty to make it 22-20.
Another moment of magic from Russell then appeared to have swung the game back in Clermont's favour as his clever break resulted in Wesley Fofana powering over underneath the uprights. But as the traveling support started to celebrate, referee Nigel Owens brought play back for a forward pass that saw Stade escape unscathed.
Russell then contrived to undo his promising work with another careless error on 65 minutes. The enigmatic playmaker failed to find Murimurivalu with a poorly-timed pass, again handing Stade a simple interception score.
This time it was left wing Guillaume Bousses who took the honours, racing through unchallenged to stretch Stade's lead to seven points.
Beauxis kicked the conversion to leave Clermont trailing by nine with 14 minutes remaining but Vern Cotter's men made it a pulsating final few moments when Cudmore crashed over with 76 minutes gone.
Crucially, Parra missed the conversion, ensuring Clermont had to turn down a late long-range penalty in favour of a kick, catch and drive.
And while they did come to within 15 metres of the Stade line with one final flurry, the hosts held firm to reach a third European final, in Cardiff in three weeks time, and their first since the Heineken Cup defeat to Toulouse in 2005.