Labour's new leader said he wanted the weekly sessions to be less "theatrical" and Mr Cameron agreed there should be more focus on "substantial issues".
Mr Corbyn asked about housing, tax credits and mental health service cuts.
The BBC's Norman Smith said it was much calmer than normal but the PM had not been put under any real pressure.
Speaking to the BBC's political editor Laura Kuenssberg later on Wednesday, Mr Corbyn insisted he would not campaign for the UK to leave the European Union, while vowing to fight any moves towards a more free-market relationship.
Mr Corbyn was cheered by Labour MPs as he got to his feet at his first PMQs, before launching into a lengthy pre-amble in which he thanked supporters for his Labour leadership victory, vowed to make PMQs less "theatrical" and explained his decision to crowd source his questions.
He said: "I thought at my first PMQs, I would do it in a slightly different way... So I sent out an email to thousands of people and asked them what questions they would like to put to the prime minister and I received 40,000 replies."
The prime minster congratulated Mr Corbyn on his "resounding victory" in the Labour leadership contest and welcomed him to the frontbench.
Mr Cameron, who began his own career as opposition leader in 2005 promising to end "Punch and Judy" politics, then said "no one would be more delighted than me" if PMQs could become a "genuine exercise in asking questions and answering questions".
Mr Corbyn's first question "from a woman called Marie" was on the "chronic lack of affordable housing" and "the extortionate rents charged by some private sector landlords" - something the new Labour leader has said will be his top priority.
Mr Cameron said the government had delivered more affordable homes but recognised "much more needs to be done".
Before Mr Corbyn began his second question, on job losses at housing associations, he thanked the prime minister "for his commitment that we are going to do Prime Minister's Questions in a more adult way than it's been done in the past".
He then asked about "absolutely shameful" cuts to tax credits voted through by MPs on Tuesday, on behalf of  "Paul," and about cuts to mental health services, prompted by a question from "Gail".
BBC political editor Laura Kuenssberg's verdict
As promised, it was different. A parliamentary revolution in beige, Jeremy Corbyn's new leader's navy suit, no where to be seen.  It wasn't so much Jeremy Corbyn's questions to the prime minister but Gail, Marie, Steven, and Angela's questions.
What's also different? Camp Corbyn and David Cameron's team are both happy with the outcome of the crucial half hour of PMQs, the new Labour leader's first ever outing at the despatch box.
Read Laura's full analysis
Mr Cameron struck a consensual tone in his replies, agreeing that more needed to be done to improve mental health services, and paused to scold Labour MPs who were jeering him over his claim that the lowest paid would be Â£20 a week better off next year thanks to an increase in the minimum wage.
"I thought this was the new Question Time. I'm not sure the message has fully got home," he told them.
The prime minister took a couple of swipes at the new Labour leader when answering questions from backbenchers, as noise levels in the chamber returned to normal levels, although Tory MPs still appeared to be following instructions to avoid triumphalism.
In response to a question on Northern Ireland from DUP MP Nigel Dodds, Mr Cameron paid tribute to former Conservative MPs Airey Neave and Ian Gow, who were murdered by Republican terrorists.
He did not directly refer to controversy over Mr Corbyn and shadow chancellor John McDonnell's previous comments on Northern Ireland, but said: "I have a simple view, which is the terrorism we faced was wrong, it was unjustifiable, the death and the killing was wrong. It was never justified and people who seek to justify it should be ashamed of themselves."
He held back from attacking Mr Corbyn over his much-criticised decision not to sing the national anthem at a service to mark the 75th anniversary of the Battle of Britain, saying in response to a question from a Tory MP that the St Paul's Cathedral event had been "a reminder of how proud we should be of our armed forces, then, today and always."
Minutes before the session began Labour sources confirmed that their leader, who believes in the abolition of the monarchy, would join in with the anthem at future official engagements.
Labour denied that Mr Corbyn's new approach - asking six different questions rather than pushing the prime minister on a single point - gave Mr Cameron an easy ride.
"The questions were very direct and they demanded clear answers," a Labour source said.
Asked about policy differences that appear to have emerged between Mr Corbyn and his front bench, on welfare cuts and Europe, a Labour source said "lots of listening" would be done and some policies would be reviewed under the new leadership.
Jeremy Corbyn asked David Cameron questions emailed to him from the public as he tried what he called "a different" style for his debut PMQs.