Fleming finished on 82 not out off 143 balls and hit 13 fours, while McMillan had 15 boundaries in his 125-ball unbeaten 85. The winning runs came an hour after lunch with the second-wicket pair scoring 88 in 16.1 overs in the afternoon session to maintain their unbeaten record against county sides.
At the start of the third day Leicestershire - on 333 for nine and a lead of 198 - looked to have an outside chance of victory if their last- wicket pair could add 30 or 40 more runs.
But those hopes were quickly ended by the left-arm seamer Shayne O'Connor who bowled James Ormond with the second ball of his first over without further addition to the overnight total. The impressive O'Connor finished with 6 for 65 to push his claims for a place in the Test side.
Although the tourists lost opener Matthew Bell - trapped lbw by left- arm spinner Matthew Brimson with the total on 32 - that proved to be their only moment of concern.
Against some loose and wayward bowling from the Leicestershire attack Fleming and McMillan helped themselves to a succession of boundaries and, more importantly, enjoyed the sort of time in the middle and batting practice they needed before the Manchester Test.
New Zealand's delighted coach, Steve Rixon, said: "There were several plus points from the match for us, but best of all was the fact that the two guys who needed batting practice the most had genuine time out there in the middle.
"They have not been filling the scorebook full of runs recently and they needed that.
"Geoff Allott also bowled well which was pleasing, and O'Connor's performance has put pressure on our top-order bowlers."
Fleming said: "I have been feeling OK but not getting the runs, so it was good to have time in the middle like this. It was also good to win the game because it is important to maintain the winning habit."
On Saturday, Northamptonshire's David Sales again dominated the day's cricket action when he became the youngest English batsman to score a triple century in first-class cricket.
The 21-year-old, who was only awarded his county cap yesterday, struck an unbeaten 303 in just under eight hours to steer Northamptonshire to 579 all out.
Sales' eight-hour occupation completed a dramatic transformation in the game as Northamptonshire had been 68 for 6 at one stage.
His was the second-highest innings in Northamptonshire's 94 years as a first-class county, and he looked to have Mal Loye's county record score of 322 not out - made off Glamorgan's attack only last season - in his sights before running out of partners.Reuse content