England, chasing a victory target of 130, lost Ben Duckett to the first ball and opening partner Emilio Gay later in the same over from Muhammad Abbas.
But Root, who had a fortunate escape when he was bowled by Mohammad Ali on 23 off a no ball, brought all his experience to bear as England wrapped up their first series win for 19 months.
It was Ali’s 17th no-ball of the series.
Pakistan’s last chance came and went when Root, on 38, edged Abbas between first and second slip, who both left the ball for each other.
It was Root’s 110th innings of 50 or more in Test cricket, a total which includes 41 hundreds.
Pakistan, who conceded a first innings deficit of 320, showed commendable fight to take the game into a fourth day, but with Jordan Cox supplying solid back-up to his captain with an unbeaten 59, England eased to victory in the 25th over.
"Fair play to Pakistan, they threw some punches last night. They made it very difficult for us, and in some ways it's a good thing to go through that," said Root.
"Test cricket is not always going to go your own way, but to see us actually be quite clinical with the bat off the back of that was good too. So in many ways we're better for it.
"Of course there are things we will want to improve on, but we came into this series with a lot of question marks and a lot of noise around the team, and we’ve walked away 3-0 winners so in that respect I’m quite pleased."
If only Pakistan had shown up in the first two Tests as they did belatedly in Birmingham, this could have been an enthralling series. As it was, England's total dominance made it something of a damp squib.
England, attempting to reboot after the New Zealand defeat which followed the Ashes loss, will not be complaining.
Notwithstanding the panic induced by Pakistan’s exciting debutant Razaullah, whose thrilling 81 from 63 balls electrified Edgbaston on Friday evening, it has been a satisfying return to the captaincy for Root following Ben Stokes’ retirement from international cricket.
Marcus Trescothick, appointed interim red-ball head coach after the sacking of Brendon McCullum, hands over a rejuvenated side to Stephen Fleming for the three-Test series in South Africa in December and January.
South Africa will be much stronger opposition, but if England can keep the quartet of Ollie Robinson, Jofra Archer, Josh Tongue and Gus Atkinson fit, they will be a threat.
"I’m a realist, I know it's not always going to be like this," said Robinson, England’s player of the series.
"We know there’s going to be tougher times ahead. We have some tough opposition coming up, and we’re looking forward to the challenge."
The England pace bowling attack, led by Robinson, who took 21 wickets, was too good for Pakistan on helpful surfaces.
The batting was not quite so convincing, with no one able to make a century. With Jacob Bethell due to come back in for the winter tour after knee surgery, one batsman out of Emilio Gay, Dan Lawrence and Cox will have to make way.
That situation remains fluid, but Cox, who also kept wicket in the second innings because of Jamie Smith’s finger injury, did his cause no harm with his poised second-innings performance.
For Pakistan, who have now lost 16 of their last 21 tests, it has been a chaotic tour capped off by the decision to replace seven players in the squad after the second test and sack coach Sarfaraz Ahmed and his assistant Umar Gul.
At least, though, the ‘Raz-ball’ fightback in Birmingham showed there was still a pulse ahead of their next assignment against Sri Lanka in October.
"We had a couple of good moments. I think we have bowled well, but I think the first innings cost us. Too many soft dismissals," said Pakistan captain Babar Azam.
"I think this was a good opportunity for youngsters to play against quality bowlers, quality opponents, so they learned a lot, and I think this is helpful for the future."