The changes will take effect from May 15.
Replacing him at the Education Ministry will be Trade and Industry Minister Chan Chun Sing, who in turn will relinquish his portfolio to Health Minister Gan Kim Yong.
Transport Minister Ong Ye Kung will take over from Gan as Health Minister, and co-chair the multi-ministry task force handling Covid-19 with Lawrence Wong.
The Cabinet reshuffle will see seven of the 15 ministries helmed by new ministers.
Communications and Information Minister S. Iswaran will be Minister for Transport, with Manpower Minister Josephine Teo replacing him at the ministry.
Second Minister for Manpower Tan See Leng will move up to helm the Ministry of Manpower in her place. He will continue as the Second Minister for Trade and Industry.
Speaking at a press conference, PM Lee noted that he is making a more extensive reshuffle than is usual this early in the term of government, which began last July after the general election.
There are two reasons for this, he said.
First, there are many repercussions with Deputy Prime Minister Heng Swee Keat relinquishing the key portfolio of finance.
Second, he is moving the ministers heading the front-line ministries dealing with Covid-19 - Health, Manpower, and Trade and Industry.
PM Lee said he had considered making these changes after the general election, but the country was still in the thick of battling Covid-19.
He added that he is now able to do so as the Covid-19 situation is more stable.
Noting that most of the fourth-generation ministers have already accumulated experience in a wide range of portfolios, he said this round of Cabinet changes will allow them to gain new experience and exposure.
The reshuffle is major but not a sudden one, PM Lee added.
"The appointments which I made immediately after the elections, in a sense, were an interim set of changes, and now I'm making the full set of changes," he said.
"And therefore I've had to move two ministers who have only been in their appointments quite a short time, which are Lawrence in MOE and Ong Ye Kung in MOT, but unfortunately it can't be helped.
"It's a bit disruptive for the ministries; they have made contributions, they are getting into their stride, and now I have to disrupt them. But I hope that after this adjustment, the new ministers in those two posts will be able to settle down for some time.”
Elaborating on the changes, PM Lee said that by relinquishing his finance portfolio, Heng will be freed up to concentrate more on the whole-of-government economic agenda, including chairing the Future Economy Council.
Heng will remain DPM and Coordinating Minister for Economic Policies, as well as oversee the Strategy Group within the Prime Minister's Office and the National Research Foundation.
He will also continue to co-chair the Joint Council for Bilateral Cooperation - the highest bilateral platform between Singapore and China - with Chinese Vice-Premier Han Zheng.
PM Lee said Mr Wong is a "natural fit" to be Finance Minister, as he has been Second Minister for Finance since 2016.
The Prime Minister said Chan "has done an excellent job getting our economy back on track, and preparing our industries and companies to respond to structural changes in the global economy".
"Now I am sending him to Education, where he will build on the work of previous education ministers to improve our education system to bring out the best in every child and student, and develop young Singaporeans for the future," he added.
"Nurturing people is quite different from growing the economy or mobilising unions. I look forward to Chun Sing taking on this fresh responsibility and broadening his experience."
Gan, who takes over as Trade and Industry Minister, will oversee Singapore's economic recovery from the pandemic and pursue new growth opportunities, said PM Lee, adding that he is well-suited for the role as he was previously Manpower Minister and had served in the Ministry for Trade and Industry early in his career.
On Ong, he said the current Transport Minister has been dealing with major challenges since he took over from Khaw Boon Wan.
"Since Covid-19, Ye Kung has been working on reopening our borders and protecting our status as an air and sea hub," PM Lee said.
"At MOH, he will build on the strong foundation that Gan Kim Yong laid in healthcare."
In other changes, Sim Ann will be appointed Senior Minister of State for Foreign Affairs, replacing Chee Hong Tat.
She will continue as the Senior Minister of State for National Development, but relinquish her appointment as Senior Minister of State for Communications and Information.
Tan Kiat How will take over as Minister of State for Communications and Information, among other new postings. - The Straits Times/ANN