Figures released by the west London airport show that 6.1 million travellers used the hub last month.
The growth was driven by emerging markets, with China up 12%, the Middle East 9%, Mexico 6% and India 4%.
Carriers have benefited from new larger aircraft such as the Airbus A380.
Garuda Indonesia launched the UK's first direct service to Jakarta in March.
Heathrow chief executive John Holland-Kaye said: "It's been a record March for Heathrow and I'm delighted that we've added the UK's first direct connection to Indonesia.
"As South East Asia's largest economy, Garuda Indonesia's new route will be a significant boon for British SMEs (small and medium-sized enterprises) looking for new export opportunities."
Some 3.1 million passengers used Gatwick last month, up 5.8% year on year.
Stansted welcomed 1.9 million passengers during March (a 13% rise), while Dubai, Dublin, Amsterdam, Tenerife and Alicante were the most popular international routes from Manchester, which experienced 11% growth to 1.7 million passengers.
Luton Airport recorded 1.1 million passengers last month, a 24% increase year on year.