e was admitted to hospital on Sunday after becoming the latest official close to Mr Trump to test positive.
Mr Giuliani, 76, told the show he expects to leave hospital on Wednesday.
He has been treated with Remdesivir and Dexamethasone, he explained.
Mr Trump tweeted on Sunday that his ally, who has been leading the Trump campaign's legal challenges to the November election outcome, had been diagnosed with the virus.
"I am doing fine. Pretty much all the symptoms are gone. The minute I took the cocktail I felt 100% better. It works very quickly, wow," he told his colleagues on his weekly show with 77 WABC radio from the Medstar Georgetown University Hospital in Washington DC.
Mr Trump has strongly praised the experimental combination of drugs he received when he spent three nights in hospital with Covid-19 in October.
Dozens of people in Mr Trump's orbit are said to have tested positive for Covid-19 since October.
media captionFour Covid rules broken by Trump and the White House
Mr Giuliani said the president's doctor had urged him to go to hospital where he could "get it [Covid-19] over with in three days".
Mr Giuliani's son Andrew tweeted that his dad had "improved significantly" adding "I can't seem to get him off the phone for the last day".
The BBC is not responsible for the content of external sites.View original tweet on Twitter
Referring to his prior diagnosis of prostate cancer, Rudy Giuliani suggested "You don't screw around your whole life because of an illness. I'd rather face risks than live in a basement my whole life."
During the election campaign earlier this year, Mr Trump's campaign attacked his rival Joe Biden for "hiding in his basement" during the pandemic.
Mr Giuliani, a former mayor of New York City, had been on a cross-country tour in an effort to convince state governments to overturn the results of the November election vote when he contracted the disease.
He had criticised face masks and was frequently pictured at indoor events without a face covering.
media captionOne travel nurse's journey to US Covid hotspots
Last Wednesday, President Trump's lawyer appeared at a hearing on alleged election fraud in Michigan where he asked a witness beside him if she would be comfortable removing her face mask.
"I don't want you to do this if you feel uncomfortable, but would you be comfortable taking your mask off, so we can hear you more clearly?" said Mr Giuliani, who was not wearing a face mask. The witness chose to keep her mask on after asking the panel if she could be heard.
The US has recorded more than 15 million cases so far and 285,000 deaths, according to Johns Hopkins University research, which are both global highs. Many parts of the country are seeing peak infections, with record numbers of people in hospital.
Correction: an earlier version of this story wrongly stated that Regeneron made Remdesevir, a drug developed by Gilead.
image captionBoeing 737 Max aircraft were grounded in March 2019
Brazil's Gol has become the first airline to resume commercial flights with the Boeing 737 Max.
The plane had been grounded worldwide since March 2019 following two deadly crashes which killed 346 people.
But following an overhaul, the first passenger flight took off from Gol's hub in São Paulo on a flight to the city of Porto Alegre.
Gol had earlier said that 140 of its pilots had undergone training on the overhauled plane in the US.
So far, only the US and Brazil have recertified the jet.
Gol, the biggest domestic airline in Brazil, has seven Boeing 737 Max planes which it plans to use on 27 commercial domestic flights.
Boeing was forced to take the 737 Max out of service following two crashes within five months of each other, which together killed 346 people.
The aircraft manufacturer has since implemented a series of modifications including updating flight control software, revising crew procedures and rerouting internal wiring.
Gol's vice-president for operations, Celso Ferrer, said he had full confidence in the plane's safety following the modifications.
"For the past 20 months, we have been carrying out the most intensive safety review in the history of commercial aviation. Safety comes first and foremost," he wrote in a statement.
media captionZipporah Kuria's father Joseph Waithaka was one of 157 people killed when a Boeing 737 Max crashed in March 2019
A company spokesman told AFP news agency that passengers could reschedule at no extra cost if they did not want to travel on the Boeing 737 Max.
"They will be able to reschedule the trip without fees or fare difference, maintaining the same origin and destination, within the validity period for the ticket, which is 12 months from the date of purchase," the spokesman said.
The resumption of commercial flights is a boost for Boeing. As well as being the first commercial airline to use the Boeing 737 Max on commercial routes, Gol also confirmed it had 95 standing orders for the 737 Max with Boeing.
The news comes less than a week after Irish airline Ryanair said it had placed an order for 75 Boeing 737 Max aircraft.
6 days ago