Johann van Graan's team will face Northampton in their first Premiership final for nine years despite a stirring fightback from Sale at the Recreation Ground.
Sale's defeat was compounded by claims of an ugly altercation between England lock Hill and a Bath supporter.
Hill, 29, is currently recovering from injury and watched the game with Sale's coaching team in the main grandstand.
As he left his seat at the end of the game, he walked down the stairs and apparently became involved with the 28-year-old fan, who told reporters Hill "grabbed me around the throat, broke my sunglasses and cut the top of my eye".
Responding to the allegation, the Sharks said: "Sale are aware of an incident involving a player and Bath Rugby fan at the final whistle of today's semi-final.
"The club will work with Bath Rugby to complete an investigation into the incident and will provide no further comment at this time."
Sale rugby director Alex Sanderson added: "It is news to me. Jonny is not the aggressive type and I will answer those questions when I know.
"I can vouch for Jonny's character. He is not an overly aggressive spiteful person."
'Belief is back'
Sale had wiped out an 18-5 deficit before two second-half Finn Russell penalties and a late Niall Annett try powered Bath into next Saturday's title match at Twickenham.
Ted Hill and Beno Obano also scored for Bath, while Russell kicked 16 points from four penalties and two conversions.
Sale also scored three tries as Ben Curry, Tommy Taylor and Tom O'Flaherty all crossed, with George Ford adding two penalties and a conversion.
There was no fairy-tale return to action for Sale and England flanker Tom Curry, who featured for 33 minutes off the bench on his first appearance since the World Cup after undergoing major hip surgery.
Hailing Bath's progress to the final as a reward for his patient rebuilding job, Van Graan said: "I'm very thankful for days like this.
"This group of players and staff have made a difference to the lives of the people of Bath. We've put belief and pride back into the jersey.
"I saw them belonging to an incredible club today and we've created a journey that's gone from hope to belief, and the belief is back in Bath Rugby."
