James Stark, 37, Steven Sheldon, 35, and Martin Williams, 36, all from south Wales, died in Wiltshire in June 2014.
Stephen Jenkins, 39, of Abercwmboi, was driving a van that crashed into a lorry, near Chippenham.
He was convicted of causing death by dangerous driving at a previous hearing at Swindon Crown Court.
Speaking after sentencing, Sgt Barrie Card, from Wiltshire Police, said: "Falling asleep at the wheel doesn't happen straight away, you will get telltale signs and that's when you must act before it's too late.
"Every driver owes it to themselves, their passengers and other road users to make sure they are fit to get behind the wheel, and that includes not being too tired.
"The road safety message 'tiredness kills' is true and sadly so true in this tragic case."
Jenkins, of Park View Terrace, was also disqualified from driving for four years and three months and will have to take an extended retest to get his driving licence back.

Brownlee, 28, won Olympic gold at London and Rio, and has not ruled out bidding for a third win at Tokyo 2020.
"I'm going to pursue things away from Olympic triathlon; it's important I have a different focus," Brownlee said.
"I've been to three Olympics and I can't say I won't definitely be at a fourth."
Ironman triathlons consist of a 2.4-mile swim, a 112-mile cycle and a full marathon - almost four times further than an Olympic triathlon.
"All triathletes have the ambition to do the almost mystical Ironman World Championships in Hawaii," added Brownlee, who beat his brother Jonny into second place at the Rio Games.
"It's very unlikely that I could do it in the next two years, but I could build up my endurance then maybe go back to it after Tokyo as a 32-year-old."
Brownlee has also revealed his mixed feelings over the attention that followed his decision to help his brother over the finish line at September's season-ending World Series event in Mexico.
Leading with 700m left, Jonny, 26, began to weave over the road in hot and humid conditions in Cozumel.
Third-placed Alistair caught his brother, propping him up for the final couple of hundred metres before pushing him over the line in second place.
"Both Jonny and I were really quite uncomfortable with the attention that came out of that day, especially while Jonny was in intensive care and the jury were deliberating on whether we should be disqualified," he said.
"The scale of the after-effect was incredible and, to be honest, completely baffling. We were both embarrassed at first and wanted to turn down all the attention, but we realised it gave us a platform to talk about triathlon and encourage people to get active.
"The whole Mexico thing brought us to the attention of a lot of people who probably didn't know what triathlon was. So I think now that we can look back on it with the benefit of hindsight, it was probably a good thing - although I'd still have preferred Jonny to have won the race."
A van driver who killed three passengers in a crash on the M4 after falling asleep at the wheel has been jailed for four-and-a-half years.
[NEXT_CONCEPT]
Double Olympic triathlon champion Alistair Brownlee is to take two years out from the sport to train for the World Ironman Championships in Hawaii.