The broadcaster, dubbed Whispering Bob because of his softly-spoken style, said it was "devastating" when he was diagnosed with the "aggressive" illness.
Appearing on today's Desert Island Discs on Radio 4, he said: "I was fortunate, very, very fortunate to catch it in time and then it was a question of arresting the growth and then treatment".
Harris, famous for presenting The Old Grey Whistle Test on TV, told host Kirsty Young that Plant helped when he started to get ill again.
He said: "Every now and again all the readings will go shooting up and that happened to me a couple of years ago when it looked as though it really got serious again and I really needed to talk to somebody about this and I phoned Robert Plant and he offers such unbelievably good advice and he really saw me through that moment and then when we got the all clear again he said 'fine, OK well I'm not going to be phoning you every single day now'".
Harris, who chose a book of cricket statistics and a greenhouse as his luxury item to take to the desert island, admitted it had been "so tough" to whittle down his record collection to just eight tracks.
He said leaving John Lennon's music out was "most difficult of all" but chose a track by Plant, a cover of Gone, Gone, Gone by the Everly Brothers, from his album Raising Sand with Alison Krauss.
His other tracks included the Ben E King single, Stand By Me, and Alone Again Or from Love's Forever Changes album which he was given by John Peel the first time they met.
Harris, who joined Radio 1 in 1970, described fellow DJ Jimmy Savile as "just somebody I didn't like".
Savile, who worked for the station for almost 20 years, has been unmasked as a serial sex-offender since his death in 2011.
Harris said: "You just didn't get a good atmosphere off him at all."