Former Leicester coach Phillips claimed the Italian – who was controversially sacked last month – lacked organisation and his methods left the players to "police themselves".
Ex-assistant boss Shakespeare has been confirmed as Ranieri's replacement as manager until the end of the season ahead of the Premier League winners' Champions League last-16 second leg against Sevilla on Tuesday.
Phillips was with Shakespeare on the coaching staff at Leicester under former manager Nigel Pearson and initially stayed on after Ranieri's arrival before leaving to join Derby County in September 2015.
"He (Shakespeare) was holding the place together," Phillips said.
"I worked there for a short period under Ranieri and at times the organisation was not good. If it wasn't for Shakey it could have fallen apart a lot earlier.
"You also have to give the players credit. They kind of policed themselves last year. They managed themselves really well.
"Of course the manager takes all the credit for winning the league - which he deserves an awful lot of - but it would have been a lot more difficult for them to win it without Shakey being there."
Phillips added: "It was certainly a learning curve for me and Shakey, because at that time it was us running it with [goalkeeper coach] Mike Stowell, and Ranieri came in.
"We would have our sessions planned for the players and literally as we were walking out onto the training pitch Ranieri would come over and say, 'I want to do this now with them, and he would change it'.
"We were like, 'Fine, go on then, you take it'. But he'd say, 'No, no, no you take it'. And he's telling us to put a session on that he wants and we don't know what it is. We had to adapt.
"Shakey can do that, he can adapt in tough situations. He's got that talent. That comes from being in the business a long time and working with experienced people."