Both panellists wrote articles attacking the other in the aftermath of the BBC show.
On his blog, comedian and campaigner Brand called the UKIP leader an "an end of the pier, end of the road, end of days politician".
Mr Farage wrote on the Independent website that Brand had been exposed as a "messiah with feet of clay".
The two men were joined on Thursday night's show by Conservative Communities Minister Penny Mordaunt, shadow international development secretary Mary Creagh, and Times columnist Camilla Cavendish.
In an often heated debate, with members of the audience yelling at each other and the panel, Brand called Mr Farage a "pound shop Enoch Powell" during a section on immigration, with the UKIP leader responding by saying he believed the UK was "overcrowded".
The barbs did not stop when filming ended, as Mr Farage said the performance of the "sometime comedian turned banker-basher" had been "limp".
He wrote: "Mr Brand will swan around and tout long-discredited, even conspiratorial, theories about the City of London, and 'who owns politicians' - but actually he had very little to give."
For his part, Brand said Mr Farage was "worse than stagnant, he is a tribute act, he is a nostalgic spasm for a Britain that never was; an infinite cricket green with no-one from the colonies to raise the game, grammar schools on every corner and shamed women breastfeeding under giant parasols".
The breastfeeding reference followed a row over Mr Farage's suggestion that some venues might ask mothers to "perhaps sit in a corner" to feed their babies.
Both men praised the audience of Question Time, which was filmed in Canterbury.
Brand, who was urged to stand for Parliament by one man in the crowd. added: "The only worthwhile sentiments, be they raging or insightful, come from the audience".
Other questions focused on the NHS, grammar schools and trust in politicians.
The programme was watched by an average of 3.5 million viewers, with the audience peaking at 4.6 million at the start.
This was nearly a million higher than the previous week's episode, but well below the 7.9 million viewers who watched ex-BNP leader Nick Griffin's appearance on the show in 2009.
The encounter also triggered plenty of debate on social media, with 106,000 tweets during the show - compared with 22,000 the week before.

Costco store manager Roger Cooper, 41, and ex-soldier David Cooper, 39, were convicted of killing Sameena Imam, 34, at Birmingham Crown Court.
Her body was found buried at an allotment in Leicester in January after she was killed with chloroform.
Ms Imam had been having a two-year affair with Roger Cooper and issued him with an ultimatum to leave his partner.
An eight-week trial heard that Roger Cooper spent at least a month plotting to kill Ms Imam - one of his three lovers - to prevent her exposing their affair.
The men bought poisonous metals, identified a shallow grave and communicated in text messages written in a Star Wars-based code before killing Ms Imam.
Roger and David Cooper remained calm and did not look at each other as they were unanimously convicted by a jury on Tuesday.
Ms Imam is believed to have been overpowered on a sofa at David Cooper's home in Leicester after being driven to the property from Coventry by his brother.
She was reported missing by family members in Essex after failing to return home for Christmas, sparking a major police inquiry.
Detective Chief Inspector Caroline Marsh said: "I would say this is the most callous offence I have ever investigated.
"Most murders that occur are spontaneous events driven by anger or jealousy, whereas in this case what we uncovered was weeks and weeks of planning to kill Sameena, numerous attempts on her life and a really unusual cause of death.
"It's been a particularly challenging case and a particularly complex investigation and we are extremely grateful to the jury, who have listened to the case and given us the verdict we've had today."
Russell Brand and Nigel Farage have continued to trade insults following their Question Time clash.
[NEXT_CONCEPT]
Two brothers accused of murdering a cash-and-carry manager from Cardiff on Christmas Eve have been found guilty.