The former Napoli boss came under intense pressure last month after heavy league defeats to Bournemouth and Manchester City, plus an FA Cup exit to Manchester United, while an improved performance in their League Cup final defeat on penalties to City was overshadowed by goalkeeper Kepa Arrizabalaga's refusal to be substituted.
Poyet pointed to the similar criticism that Guardiola faced when he first arrived in England as evidence that the Italian deserves to be given time to implement his style.
“I think what Sarri set out to do at the start of the season was excellent,” he told Goal.
“Most people in England were praising how the club was transitioning into a new system of playing.
“Chelsea were challenging the top teams at the start of the season but then obviously the opposition were able to analyse their new way of playing, and then it's up to Chelsea to figure out how to counteract the opposition tactics.
“It’s true that it’s been more difficult for them recently, but we really put the emphasis on the negatives.
“For example, the final against Manchester City, everyone was talking about the Kepa situation and everyone forgot that Chelsea played a very, very good game. That was on the second page of the newspaper – the front page was Kepa. And that reflects on where football is today.”
He continued: “It would be nice to see what many pundits were saying about Guardiola two years ago.
“‘There’s no chance they’re going to play like that.’ ‘There’s no chance you’re going to win the title.’ ‘He thinks you can play the same as he did in Spain, you can’t do that in England.’
“Then he wins the Premier League and everybody disappears.”
Former Real Madrid manager Zinedine Zidane, who won three consecutive Champions League titles with Los Blancos, has been linked with the post in London – but Poyet believes Chelsea should back the man they have in charge.
“If you look around now for who is the biggest manager without a club, it’s probably Zidane – he's won the last three Champions Leagues,” he said.
“But I’d rather Sarri would stay to be honest. And, first and foremost, for him to be convinced that it’s the right club for him. We sometimes forget about the coach’s point of view.
“Sarri knows the players he has. It’s a big advantage – people don’t realise how important it is when you already know the players: you know their reactions, know their characters, how strong they are physically and mentally. It's a big advantage over going through that process with a new manager who may not be able to sign new players.
"We don’t know if he will have the time, but I want him to finish the season on a high and to have everything set up for next year to be able to be better."
