O'Grady dedicated the final hour of his Sunday show to the entertainer, who died last week following a stroke at her Spanish villa.
He told his listeners: "We lost the queen of showbiz herself and I lost a very good friend."
The presenter appeared to be overcome with emotion after playing the star's 1966 hit Alfie.
"That always does me in that one - completely finishes me off," he said.
O'Grady joked about Black's wish to be remembered as a singer rather than a presenter, saying: "She said to me 'I want singer on my headstone', and I said, 'Why do you want to be known for being a sewing machine?'"
The presenter played a host of songs by Black, including You're My World and a duet with her friend Cliff Richard.
Since her death on 2 August, Black has made a posthumous comeback to the UK music charts with her 1983 compilation album The Very Best Of... reaching number 14 in the album chart.
O'Grady added: "Cilla is back in the chart and she would be absolutely thrilled - she would be modest about it, but she would be thrilled."
Black's body was flown back to the UK, her publicist confirmed on Friday but no funeral details have yet been released.
BBC Radio 2 presenter Paul O'Grady has choked up as he paid tribute to his "good friend" Cilla Black.