Vowing to build on his "legacy", she said she wanted to pay tribute to Mr Gove's achievements in the role.
The newly promoted minister, who took over in last week's reshuffle, told the Commons: "Let me be absolutely clear that I share with him total commitment to creating an education system that enables young people, regardless of their background, to unlock every ounce of their potential.
"I believe he will be remembered as one of the great reforming secretaries of state for education."
Shadow education secretary Tristram Hunt said he too wished to pay tribute, calling Mr Gove a man "full of ideas", but adding: "They just happened to be the wrong ones."
He went on: "After no change on AS levels, work experience, free schools, can you explain to the House why you are also continuing with the flawed and unpopular policy of increasing the number of unqualified teachers in our schools?
"When will you make the break and put the interests of parents and pupils above those of Tory party ideology?"
Ms Morgan hit back: "The theatrics are typical of somebody who took part in the Cambridge Footlights when he was there.
"I'm not going to take lessons from you. Oh no, wait a minute, in fact you do give lessons as an unqualified teacher. Don't you?"
Mr Hunt moved on to another question, asking whether Ms Morgan was "fully satisfied" that the Government's changes would not compromise "fairness and consistency" as pupils receive their exam results next month.
He said: "The Government's rushed curriculum changes risk undermining faith in the examination system, causing confusion for parents and pupils.
"Already Ofqual has warned of greater than normal turbulence in exam results this summer."
Ms Morgan replied: "I would like to answer that question with a one-word answer. Yes (I am fully satisfied).
"I'm not going to take lessons from you. Because it is under this Government that there are a quarter of a million fewer pupils in under-performing schools, 800,000 more pupils who are in schools rated good and outstanding.
"That is the legacy left by ... the member for Surrey Heath (Mr Gove) which I intend to be building on."
She was quizzed again on unqualified teachers by Labour's Jon Ashworth (Leicester South), who asked: "Many parents ... don't think it unreasonable to expect teachers to be qualified. Why do you disagree with them?"
Ms Morgan replied: "I think you ought to ask your own shadow education secretary.
"On this side of the House we believe that head teachers are the best people to know about the qualifications of the people who are teaching their children."
Ms Morgan's first response in the session was to a question from Labour's Jon Ashworth (Leicester South) on childcare.
The Education Secretary told MPs: "One of the greatest achievements of my predecessor (Mr Gove) was to put in place real reforms which ensure all our young people from the earliest age have the opportunity to succeed, and it is a privilege to follow him in this role."
Shadow childcare minister Lucy Powell pressed Ms Morgan about whether she would be able to devote enough time to the issue.
Ms Powell asked: "Can I take it from you answering the questions that you are now the childcare minister alongside your role as secretary of state, and despite your expanding ministerial team you have taken on those responsibilities as well?
"I'm all for flexi-working but given the challenge our country faces with the childcare system I hope you are able to focus full-time on this issue."
Ms Morgan replied: "You will be aware, as a fellow working mum, that women are excellent at multi-tasking.
"Yes, of course, as secretary of state for education I'm interested in childcare and a whole range of issues my department will be dealing with, as well as my brief as minister for women and equalities.
"I look forward to continuing these debates with you as well as the (education minister Sam Gyimah) who will also be taking on considerable responsibility for childcare."