Root could not resume batting at the start of the final day, but came to the crease an hour in when Moeen Ali was dismissed and batted to lunch.
"He was up all night with some sort of stomach bug, some gastro thing. He is asleep in the dressing-room trying to recover," Anderson told reporters.
"He's not had any sleep, he's not eaten. He's had diarrhoea and vomiting, so I'm guessing he's not in a great state at the minute," the paceman added.
"So to get to the ground in itself is a great effort and then to strap his pads on and bat for as long as he did was a brilliant effort from him and just shows what sort of character he is."
Veteran Anderson praised Root's leadership on a troubled tour plagued by off-field distractions and the unavailability of suspended star allrounder Ben Stokes.
"He wants to lead by example for this team. He's been a fantastic captain throughout this tour," Anderson said.
"You ask any captain that has toured Australia it is not an easy place to come, not an easy place to play, especially when you're on the wrong end of results."
Root finished retired ill on 58 for his fifth half- century of the series, but crucially he was not able to convert any into a Test ton.
He finished the series with 378 runs in nine innings at 47.25, the highest average of his team.
"It took a lot of courage to come out and bat, you could see he was struggling a fair bit, particularly when he was running between wickets," Smith said.
"He's a tough character so he must have been in a pretty bad way not to come out after the break.
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