Casadesus put the Balearic Islanders ahead in the eighth minute at the Iberostar Estadi before Arizmendi, a former Valencia player, doubled their lead 10 minutes into the second half moments after coming off the bench.
Valencia had their chances but Mallorca's Israel goalkeeper Dudu Aouate was on superb form and produced several fine saves to deny the visitors.
The win lifted Mallorca to second on 11 points, four behind Barcelona, whose 2-0 success at home to Granada on Saturday maintained the leaders' winning start.
Valencia, who lost 2-1 at Bayern Munich in their Champions League Group F opener on Wednesday, are 13th on five points.
Real Madrid are looking to get their La Liga campaign back on track when the champions play at city neighbours Rayo Vallecano later on Sunday.
Jose Mourinho's side have lost twice in their opening four games, the same number of defeats as in the whole of last season, and are 11 points behind arch rivals Barca.
Europa League champions Atletico Madrid host Real Valladolid and Malaga can climb above Mallorca into second with a win at struggling Athletic Bilbao.