Ward produced a stunning opening set, holding firm on serve and breaking Anderson to seize the ascendancy in their match.
But the South African responded with a crushing display of groundstrokes in the second to level up the match.
Ward battled hard in the deciding set as he targeted an early upset, but his campaign was cut short as Anderson won through to the second round.
Next up for the ninth seed will be a clash with Sergiy Stakhovsky who defeated Flavio Cipolla 6-4 6-1.
Britain’s Jamie Baker beat compatriot Oliver Golding 7-6 6-3 to set up a daunting second-round clash with second seed Jo-Wilfried Tsonga.
Baker, who should break into the Top 200 for the first time after reaching round two, is looking forward to the opportunity of facing a man he watched on TV last week.
“I have never played somebody ranked as highly as that in a competitive match,” said Baker of the world number five.
“I was actually watching him play in the quarters (of the French Open) last week and I said to the person I was watching with, ‘I wonder what I'd do if I played one of these guys.’ I guess I'm going to have to think about that tonight!”
Tsonga received a bye through the first round, along with top seed and defending champion Andy Murray, who faces the 2007 finalist Nicolas Mahut in round two. Mahut defeated Guillemo Garcia-Lopez 6-4 6-2 on Tuesday.
Gilles Muller, the 14th seed, beat another Brit in the shape of Liam Broady with a very comfortable 6-2 6-1 victory.
Muller cruised past Broady in the opening set, breaking him at will in a hugely dominant display.
Broady was unable to find any response in the second set as he won just three games all match.
Next up for Muller will be a second-round match with Grigor Dimitrov who beat qualifier Bobby Reynolds of the USA 6-3 7-5.
After rain cleared away around 1pm, virtually all of the scheduled matches were completed.
Although unseeded, perhaps the biggest upset of the day was the demise of the four-time champion Lleyton Hewitt, who went down 6-3 6-2 to Ivo Karlovic of Croatia.
Elsewhere, there were wins for Marcos Baghdatis and David Nalbandian.
On Wednesday, another four-time winner, Andy Roddick will open the action on Centre Court against Edouard Roger-Vasselin of France, with Murray vs. Mahut, Tsonga vs. Baker, and then third seed Janko Tipsarevic against Ryan Sweeting to follow.