Full-back Packman dotted down twice early on and there were also scores for Harry Mallinder, George Cox and George Furbank, with Mallinder kicking six points.
Saints and Exeter had finished top of the Northern Conference and Southern Conference respectively, earning a shot at glory in the last game of finals day.
And it was to be the Northampton youngsters who prevailed thanks to a superb performance.
Packman got things off to a flying start with two tries to put his side clear and Mallinder added a third, while kicking well to take his team into a 19-0 lead.
Exeter responded with a try either side of the break, but Cox ensured the Northampton young guns retained breathing space.
Replacement Furbank extended the lead before Exeter grabbed a late consolation.
But Saints had done enough to ensure the silverware would be heading back to Northampton.
Saints: Howard Packman; Tristan Rawcliffe, Rory Hutchison, Harry Mallinder (c), Rotimi Segun; Harry Sheppard, Michael Morris; James Fish, Owen Griffiths, Luc Jeannot, Jordan Onojaife, Josh Peters, George Cox, Lewis Ludlam, Ollie Lyons.
Replacements: Freddie Davison, Harrison Marshall, Jack Strong, Karl Wilkins, Will Allman, George Chatterton, Nico Defeo, George Furbank.