The UK has 34 universities in the Times Higher Education ranking of the world's top 200 institutions.
Oxford climbs to second place, while Cambridge and Imperial College London also make the top 10 in fourth and eighth places respectively.
Keeping the top spot for the fifth consecutive year is the California Institute of Technology in the US.
Europe has a record number of universities in the world top 200, with 105 compared to 87 last year.
The tables rank universities worldwide on measures like teaching, research and international outlook - for example numbers of overseas students and staff.
The majority of UK universities have moved up this year, some - for example Warwick, St Andrews and Exeter - by a significant margin.
There is good news for Reading, Dundee and Newcastle, re-establishing their places in the top 200 after slipping out last year.
However, four universities - Manchester, York, Sussex and Royal Holloway, London - have slipped to lower positions in the tables, compared with last year when there were only 29 UK institutions in the top 200.
While the US remains the world leader when it comes to elite universities, its dominance has been eroded this year.
It has six of the top 10 universities - down from seven last year - and 39 of the top 100 - down from 45 last year.
There is a mixed picture for Asia, with Japan and South Korea falling back this year and China remaining steady.
Europe is catching up on the dominance of the Anglo-American universities, with the Swiss Federal Institute of Technology, Zurich being the first institution from outside the US and UK to make the world top 10 in a decade.
Germany has 20 universities in the top 200 and the Netherlands has 12 and there are five from France, while Spain and Italy each have three.
Phil Baty, editor of the Times Higher Education world university rankings, said: "The UK is a stand-out performer in this year's rankings, boasting an impressive 78 institutions overall, with 34 of these sitting in the top 200.
"However, despite the UK's success, its continued cuts in higher education funding - the Higher Education Funding Council for England received a £150 million budget slash this year - and series of immigration measures affecting overseas students, will hinder its performance in the long run.
"Many of the country's European rivals, such as Germany, Switzerland and the Netherlands, are also performing well, but are less hindered by funding cuts and more welcoming for international students.
"The UK will have to work hard to ensure its higher education spending and immigration policies do not hinder its place in the world university rankings."
Dr Wendy Piatt, director general of the Russell Group of leading UK universities, said "The UK spends much less on higher education and research than our nearest rivals.
"Our competitors in China, Germany and Japan continue to be rewarded with significant investment and are snapping at our heels as a result."
Nicola Dandridge, chief executive of Universities UK, said: "If we want to maintain this leading position, we must start matching our competitors' increased investment in higher education.
"We should also be presenting a welcoming climate for genuine international students and academics and ensuring that visa and immigration rules and procedures are proportionate."
Universities Minister Jo Johnson said: "It is great to see the UK is second only to the US for the number of world-class universities in the top 80.
"These rankings confirm the world-class standing of our higher education sector. Our reforms will ensure our universities continue to compete with the very best internationally and deliver high-quality teaching to students at home."