Root became England's leading run scorer in the longest format on his way to a 35th Test century on day three of the first Test against Pakistan in Multan on Wednesday.
When Root reached 71 before lunch he passed the previous milestone of 12,472 runs set by his former captain Cook and moved to fifth on the all-time list.
Indian great Tendulkar tops the chart on 15,921 runs, but Cook said that Root, who is 33, has plenty more years left at the top.
"I can see him overhauling Sachin Tendulkar's record," Cook said while commentating on BBC radio.
"You could say Sachin is still the favourite but just.
"I don't see that happening for Root to lose that hunger and ability to keep driving himself forward for the next couple of years."
Root's achievement was hailed by current England captain Stokes, who is sitting out the first Test with an injury.
"The selflessness that he has is an incredible attribute for him," Stokes said in a video on the England and Wales Cricket Board's social media channels.
"He always puts the team first, and the fact that he's got so many runs is just a bonus for us. He's an incredible player."
Two other former England captains, Michael Atherton and Nasser Hussain, joined those paying tribute to Root, who made his Test debut in Nagpur against India in 2012.
"Twelve years of excellence is exactly what it has been," said Atherton during television commentary.
"I was there in Nagpur, I thought: 'This guy's going to be one of our greats,' But you've still got to do that."
Co-commentator Hussain echoed the praise.
"He has graced us with such amazing ability and shots and temperament and hunger, and throughout those 12 years he's played with a smile on his face, which we all know is not easy," said Hussain.
British Prime Minister Keir Starmer sent Root a message on X: "Congratulations to Joe Root on becoming England's leading Test run scorer. A fantastic achievement from an outstanding cricketer who just gets better and better."
