Nine-man Liverpool lost after Joel Matip scored an own goal in stoppage time but they should have taken the lead in the first half through Diaz whose goal was ruled out for offside.
VAR failed to overturn the incorrect decision to disallow the goal leading the English referees' body PGMOL to issue a statement blaming "significant human error" in the Premier League game.
"It is clear that the rules of the game were not applied correctly, which undermined sporting integrity," Liverpool said in a statement on Sunday.
"It is therefore unsatisfactory that sufficient time was not given to allow the correct decision to be made and that there was no subsequent intervention."
The PGMOL said disallowing the Diaz goal was a "clear and obvious factual error and should have resulted in the goal being awarded through VAR intervention."
PGMOL added that it will conduct a full review into the circumstances which led to the error.
Dale Johnson explains the sequence of events that led to Luis Díaz's goal being incorrectly disallowed vs. Tottenham.
Liverpool's statement added: "That such failings have already been categorised as 'significant human error' is also unacceptable. Any and all outcomes should be established only by the review and with full transparency.
"This is vital for the reliability of future decision-making as it applies to all clubs with learnings being used to make improvements to processes in order to ensure this kind of situation cannot occur again."
Liverpool said the club recognised the pressures referees are under but that the application of VAR technology "should have alleviated them, not exacerbated them", adding that they will explore the options available for resolution.
"With it being the only Premier League game of the day, the PGMOL needed a real clean day and unfortunately they have got the opposite," said Cooper.
"We all want referees to be at the right level and help them but they have to help themselves. The last thing they needed was talking points about refereeing decisions."
Brentford manager Thomas Frank was unhappy his side's penalty claims were not backed up by VAR, particularly an incident in the second half when Forest goalkeeper Matt Turner escaped punishment for a challenge on forward Yoane Wissa.
"The two handballs, some managers would claim them as penalties, I won't, but I would really hate it if they were given against me," he said.
"The Wissa one is a clear penalty though. You can't go through the man, so unfortunately that's a mistake from VAR."