Gillespie, touted as a potential successor to current Test coach Darren Lehmann, said their performance in the recent series defeat to South Africa was a rare bright spot.
They finished as the first and third-highest wicket takers with Hazlewood grabbing 17 at an average of 22 and Starc 14 wickets at 30. The duo have now played 17 Tests together since Hazlewood's debut two years ago and have taken 161 wickets alongside each other.
"Starc and Hazlewood, without doubt, will be the best fast bowling combination Australia has ever produced," Gillespie, also seen a possible replacement for Rod Marsh as chairman of selectors, told cricket.com.au on Thursday.
"They'll stay fit, they'll stay healthy, they're learning their trade all the time, they complement each other very well.
"Starc (is a) really fast, aggressive, nose-and-toes attacking option who can swing the ball late at pace.
"And then you've got Hazlewood who is an aggressive bowler but in a different way. He's aggressive with his relentless line and length, he's very disciplined."
For his prediction to come true, they will have to surpass some of the greats of the game, including Gillespie who teamed up with Glenn McGrath to take 484 wickets between them in 58 Tests across a 10-year period.
Other great partnerships included Merv Hughes and Craig McDermott and Dennis Lillee and Jeff Thomson.
"In many years' time when we're reflecting on great Australian cricketers, those two (Starc and Hazlewood) will be right up there," said Gillespie, now back in Australia after a hugely successful five-year spell coaching English county Yorkshire.