The Australian Open champion won 6-3 6-3 6-4, conserving energy for a possible three matches in three days or five in seven should he reach his first Wimbledon final.
The Istomin encounter was originally scheduled for Saturday but it was cancelled due to rain.
Wawrinka's coach Magnus Norman said playing five best-of-five set matches in seven days was "not humanly possible" and the 29-year-old Swiss player criticised the lack of consultation from the organisers.
"They just say what's going to be the schedule and that's it," he told reporters.
"Even if you want to talk to them, they're not going to change anything. They don't listen to the player, they just do what they think is good for them.
"Normally if the player starts to talk to the referee they never change the decision."
ELEGANT BACKHAND
Wawrinka dominated the rallies with his superior groundstrokes, particularly his elegant one-handed backhand which looked in fine fettle.
He blazed away with 33 winners and converted four of his five break points as he reached the fourth round for the first time since 2009.
Wawrinka will next meet Feliciano Lopez for a place in the quarter-finals after the Spaniard beat American John Isner in four sets earlier on Monday.
"Me or Lopez have to play three matches in three days," said the Swiss. "It's terrible for the body.
"I was expecting them to move matches, move maybe the juniors or doubles."
Roger Federer's victory over Santiago Giraldo was the final match on Centre Court on Saturday, finishing at 7.13pm local time and leaving almost four hours to complete a match should it have started.
Wawrinka said he was surprised the Centre Court was not used to minimise disruption but praised the consistency of the officials.
"It was a tough decision to put one match (on Centre Court) because the good thing, if one thing they did well, they postponed my match and Isner v Lopez. They had to play both matches at the same time."
(editing by Josh Reich)
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