Elphick, 55, was best known for 1980s series Boon, but had also had a string of starring roles on television and character roles in films.
The star, who first rose to prominence as the eponymous German anti-hero Private Schultz, collapsed on 7 September and later died in hospital. He had battled a drink problem during parts of his 30-year career.
The service, held on the eve of what would have been his 56th birthday, included a rendition of The Great Pretender.
Morrissey, who was starred with Elphick in the hit series Boon, and was joined by actors Malcolm Tierney and Londons Burning star James Hazeldine.
Cult comedy
Afterwards, mourners went to a wake at the nearby Nags Head pub.
Elphick, was probably best-known for his role in Boon as a retired fireman turned private detective, where he starred opposite Morrissey and David Daker.
But he also played a number of film roles, including his memorable turn as Jake the poacher in cult comedy Withnail and I.
His last major role was as Harry Slater in EastEnders, with his character leaving the square after it was discovered he had had a daughter, Zoe, by his own niece, Kat Slater.