The Italian, who was brought in as the Portuguese tactician's assistant last summer, stepped into the breach earlier this month following the 34-year-old's sacking.
Since his appointment, the West London outfit have returned to form with three wins out of three in all competitions, including a 4-1 win over Napoli which saw them progress to the last eight of the Champions League.
And Gullit believes the 41-year-old has galvanised the dressing room, backing the ex-West Bromwich Albion boss to remain at the helm beyond the summer.
"[The board] have got to look at how he performs but I am happy at the way he has handled himself," Gullit told Sky Sports News.
"He has comforted the most influential players; he rested some of them in the FA Cup because other games were more important, but he gave them confidence by saying that he needed them.
"I think Robbie [Roberto Di Matteo] is doing well. He and [assistant manager] Eddie Newton are Chelsea boys and it looks like [Chelsea] have that warm feeling again.
"I hope they can continue that until the end of the season and hopefully the board will decide that Robbie can stay in charge of Chelsea."
Former PSV Eindhoven and AC Milan ace Gullit believes the experience of John Terry, Frank Lampard and Didier Drogba, all of whom netted along with Branislav Ivanovic, in the 4-1 win over the Serie A side still has a pivotal role to play for Chelsea, despite often being overlooked by Villas-Boas during his seven-month stint at the club.
"The old players showed passion and that they care about what they are doing," he added.
"There was a debate about how good they were but on Wednesday they said: 'We are still here and you still need us'. They showed up when it mattered."
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