New figures show that in the first three quarters of 2013-14, a total of 19,124 people began a modern apprenticeship (MA) - 76.5% of the Government's target of having 25,000 apprentices taken on each year.
Youth Employment Minister Angela Constance said the figures from the training body Skills Development Scotland (SDS) were "encouraging".
But she urged more employers to take on apprentices and young people, saying: "In 2014 I hope to see more employers making young people their business.
"Taking on a keen, enthusiastic young person isn't just the right thing to do, it's good for business.
"I want more employers to not just help the younger generation fulfil their potential, but harness their skills for Scotland's wider economic benefit."
Of those who started a modern apprenticeship in the first three quarters of the year, 11,643 (61%) were male and 7,481 (39%) were female.
Just over half (52%) of those who began a training place were aged 16 to 19, while the sector with the largest number of apprentices starting was construction and the related industries, with 17% of new recruits.
There were 360 modern apprentices who were made redundant in the period April to December last year - 1% of the total number in training as part of the scheme.
But over the same period, 172 were able to continue their training thanks to the Adopt an Apprentice scheme, where firms take on apprentices who have been made redundant.
Ms Constance said the figures on apprentices were "very encouraging, painting an improving picture for modern apprenticeships in Scotland".
She added: "The Scottish Government is committed to a target of at least 25,000 new starts each year and today's publication shows we are well on course to hit this for a third consecutive year, which is a credit to SDS, employers and training providers.
"But this isn't just about the numbers, there are thousands of stories underpinning such statistics, and each MA completed successfully is fantastic news, not just for the individual behind it but also for their employer and the sector they work in.
"Key areas, such as construction and engineering, are very well represented, with exciting longer-term careers available across the board."
Skills Development Scotland chief executive Damien Yeates also welcomed the apprenticeship statistics, saying: "The figures are encouraging and are a testament to the efforts of employers and individual apprentices across the country.
"Businesses that employ modern apprentices, and apprentices themselves, play a vital and valuable role in Scotland's economy.
"Skills Development Scotland will continue to support employers to provide more opportunities to help people develop their careers in the future."
Elise Littlejohn, from Edinburgh, said the craft apprenticeship she completed last year at defence electronics company Selex ES was a great career move.
She said: "Four years of experience on the job and gaining qualifications has been invaluable."
Arman Hussain, 20, from Inverkip, chose to start an accounting apprenticeship with Morgan Stanley rather than go to university.
He said: "I wanted to go into the business world straight from school and while I could have gone to university, I knew that it wasn't what I wanted to do.
"Not even the best universities could offer the work experience and skills that I developed through my apprenticeship. The modern apprenticeship has been an alternative career route to achieve the same goals had I gone to university."