The pair surpassed the previous best of 163 as the home side were bowled out for 496, a first-innings lead of 39.
Root hit an unbeaten 154 and number 11 Anderson fell for a career-best 81, his maiden test half-century.
"For Jimmy to bat like that was unbelievable," Root told Sky Sports television. "We are now in a really good position and hopefully the bowlers can back that up now."
Australia's Phil Hughes and Ashton Agar shared the previous record last-wicket partnership, ironically on the same Trent Bridge ground in last year's Ashes series in England.
Root and Anderson's stand also meant it was the first time in history that the last-wicket partnerships of both teams had reached triple figures, following Bhuvneshwar Kumar and Mohammed Shami's alliance of 111 on Thursday.
(Reporting by Sam Holden, editing by Tony Jimenez)
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