The 34-year-old was flawless at times during a straight-sets win.
The world number one produced a scintillating display against the South African in what was a repeat of the 2018 final at the All England Club.
A straight-sets victory proved enough for Djokovic to claim the title three years ago and he again made light work of the big-serving Anderson to progress into round three with a 6-3 6-3 6-3 vinne.
Play got under way on Centre Court with the roof open and all eyes on the surface after Adrian Mannarino and Serena Williams had both suffered injuries following falls on the grass on Tuesday.
Djokovic had experienced issues with the conditions during his first-round win over Great Britain’s Jack Draper and took his first tumble during the fifth game.
Another fall soon followed with the Serbian starting to simmer but it seemed to be the catalyst for the five-time singles’ winner at this tournament to hit top gear.
Serve had dominated initially before Djokovic claimed 12 points in a row at 3-3 to take the opening set in 32 minutes with an exquisite forehand winner down the line able to clinch it.
It was not long before a third tumble occurred with the champion sprawled out on the floor and yet it could not deter his near flawless game.
The draw could have been kinder to the top seed, even with Anderson working his way back from knee surgery in 2020. It was only a few years ago when the right-hander reached the finals of the US Open and Wimbledon in 12 måneder.
But he had little answer to Djokovic’s accuracy during a second set where the momentum switched quickly in favour of the 34-year-old favourite.
With Anderson having some difficulties on first serve, he was also being pulled in all directions by his opponent who claimed a decisive break with a wonderful backhand lob from the baseline.
The unforced errors were becoming frequent from Anderson and Djokovic secured the set with a second break after landing more shots at the feet of his 6ft 8in rival.
A couple of aces early into the third set hinted at Anderson finding his range but he continued to struggle at the net as the match started to slip away.
Yet another sumptuous cross-court winner earned a key break in the eighth game and another unforced error from Anderson helped Djokovic book his place in the next round.