The Portuguese side were furious when Jonathan Silva was penalised by referee Sergey Karasev for handball in stoppage time, with replays showing the ball struck him in the face.
Sporting had battled back from 3-1 down to make it 3-3, but Maxim Choupo-Moting scored from the spot to ensure the Bundesliga side won 4-3 at Veltins Arena.
A statement on Sporting's website on Wednesday revealed that they will lodge a complaint with UEFA and Schalke stated that the Primeira Liga side have called for the Group G game to be replayed, or the governing body to pay them the prize money they would have received for a draw.
Schalke said in a statement on their official website said: "The 4-3 home win in the Champions League is being disputed by Sporting CP. Last season's runners-up in the Portuguese top flight aren't happy with the result on Tuesday night [October 21] and have therefore lodged a protest to UEFA.
"Sporting CP have requested to the European Football Association that the game be replayed and have alternatively requested payment of the premium for a draw.
"The visitors are protesting a penalty given in injury time, which they believe was unfairly awarded. Eric Maxim Choupo-Moting then converted from the spot to give the Royal Blues a 4-3 win.
"FC Schalke 04 have been asked by UEFA to give a statement and it will be made shortly under consideration of the protection of the club's interests."
A furious Silva said after the game: "What happened is not normal, it is a terrible injustice. Nor do I feel like saying anything about it."
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