The 36-year-old downplayed his personal achievement, saying he was more focused on helping England win the final match of the series.
England wrapped up victory at the Oval on Tuesday, winning the final match of the series against the number one side in the world by 118 runs
Root described the feats of Cook, who made 147 in the second innings of a 118-run win, and Anderson, as "fairytale stuff".
Anderson, who now has 564 scalps, said he was happy his friend was on the field to witness his record.
India were dismissed for 345 on the fifth and final day, having been set a mammoth 464 for victory.
James Anderson removed Mohammad Shami to go past McGrath's tally of 563 wickets
The former England captain, about to become a father for the third time, said it was difficult to put his achievement into words
Cook made 147 and Joe Root, his successor as England captain, 125 as the hosts piled up 423 for eight declared in their second innings
The 33-year-old opener reached the landmark in unusual fashion when an overthrow from Jasprit Bumrah that went to the boundary