Netherlands left-back Van Aanholt, 26, was replaced by Jason Denayer just 15 minutes before kick-off.
“He had the statutory checks with the cardiologists,” said Moyes.
“The FA panel of cardiologists looked at it and advised us really late on Sunday that he should not play until further tests had taken place.”
Moyes refused to elaborate on Van Aanholt’s late withdrawal when first asked about it following Sunday’s game.
However, speaking before Wednesday’s EFL Cup match at Queens Park Rangers, the Scot said the Dutchman had since had more tests and would travel with the squad for the game at Loftus Road (19:45 BST) after being cleared to play.
“We are just surprised the FA panel didn’t contact me or my doctor until 3.55pm on Sunday afternoon,” added Moyes.
“I have never in my career been put in a situation like that. It was completely outside our control.
“We knew we had to make the correct decision for the player and for the football club.”
The Football Association said clubs could receive advice from expert members of its panel, but added the governing body had not directly advised Sunderland.
“This cardiac screen was not conducted as part of the FA’s cardiac screening programme. Therefore the FA had no involvement in the player’s referral, the screening process or the communication of the results with the club,” it said.
“On occasion, clubs may seek independent advice on a player’s cardiac health from an expert cardiologist who sits on the FA consensus panel, which is an expert body in this area.”