Mullets through to meet Horsham
Jason Wimbleton had the first chance of the game but his shot on 21 minutes was cleared off the line by Sam Lampard.
Five minutes later, Mullets keeper Ben O'Connor made a great save after defender Lloyd Walker deflected John Boswell's cross towards goal.
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Hide AdOn 30 minutes, Arundel took the lead. Rob Grove powered home an unstoppable shot from 18 yards.
O'Connor then produced another outstanding save from another team mate when Dave Towers headed towards his own goal.
Both sides missed chances before Arundel went 2-0 ahead on 68 minutes. Andy Boxall headed home Josh Sutcliffe's cross from 10 yards.
Molesey found a reply with two minutes left when Jason Wimbleton deflected a free kick into his own goal but Arundel held on.
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Hide AdArundel assistant-manager Simon Butler said: "It was a very good team performance but Andy Boxall was outstanding. We've got good strength in depth this year and have a lot of competition for places."
Mullets were without the injured Matt Huckett, unavailable Alex Duncan and suspended Chris Hibberd.
ARUNDEL: O'Connor; L.Walker, Hack, Wimbleton, Hall, Sutcliffe; Grove, D.Towers, Mike Huckett; Boxall, Tipper. Subs: Axell, Gaskin (D.Towers, Tipper 74), S.Towers (Boxall 81), Colbourne.