Big-serving Berrettini rolled over the world No 29 from Holland 6-4 6-3 in 90 minutes to reach the final of the cinch Championships.
The Italian was then propositioned by an admirer in the stands and said, smiling: “Let me think about it.”
He has won his last eight matches on grass and has now become only the fourth man to reach four consecutive finals on the surface, along with Murray, Roger Federer and Novak Djokovic.
“I didn’t know that,” he said. “It feels nice. Obviously I can’t compare myself to those three players – they made history in our sport, but I’m trying to learn from them and hopefully one day I’m going to beat them.”
On this form, Berrettini will be a major threat at SW19 once again.
The 26-year-old from Rome dropped his serve for only the fourth time this week as Van De Zandschulp hauled back an early break to trail 4-5.
But the underdog, who at one point voiced his frustration at the noise coming from the chattering classes in front of the clubhouse, was undone by a pair of Berrettini volleys as the second seed wrapped up the first set.
Van De Zandschulp’s mood did not improve as a rain delay arrived midway through the second set with him facing a break point, the 26-year-old Dutchman arguing in vain that the game should not have started because it was already raining.
Almost inevitably upon the resumption, he lost the point, sending a forehand long and shrugging in exasperation as the match slipped away.
Berrettini added: “It feels unbelievable. I’m really happy – today was a really tough match.
“It was windy again today, but I think I played my best match of the week and I’m really looking forward to the final.
“I guess this is the UK. Yesterday felt like Italy, today is more London weather. But we have to adjust to everything. I was ready when I had to be ready.”
Cilic seemed to struggle in the breezy, chilly conditions, which were in stark contrast to Friday’s sweltering heat.
Amazingly, Krajinovic had never won an ATP Tour match on grass before he arrived in west London this week.
The 30-year-old world No 48 said: “It’s an unbelievable feeling. I just can’t believe I’m in the final. I was coming here maybe just to play one match, maybe two – now I’m in the final. It’s a dream come true.
“To be honest I don’t like to play on grass. Every year I find a way to skip the tournaments, just coming to Wimbledon and losing in the first round.
“But maybe it will change and I’ll start to like it. I have nothing to lose and I’m going to enjoy it.”
There was a surprise in the wheelchair tournament, with British top seed Alfie Hewett losing in three sets to Joachim Gérard of Belgium.
Meanwhile, at the Halle Open in Germany, world number one Daniil Medvedev will face fifth seed Hubert Hurkacz of Poland in Sunday’s final.
Hurkacz won a thriller against Australian maverick Nick Kyrgios via a third-set tie-break.