Djokovic beat seventh seed Berdych 7-6 (7/5) 6-4 6-3 to move into his 13th consecutive grand slam semi-final.
Watching among the bumper crowd was Djokovic's fellow Serbian Vidic, who had left his seat vacant in the Royal Box on Centre Court to watch his countryman play on Court One.
Having come through his toughest test of Wimbledon, Djokovic will beckon Vidic back to south-west London for his semi-final against Juan Martin del Potro.
"He's a lucky charm. I'll make sure he's there (for the semi-final)," the 2011 champion said.
"We wanted to invite Nemanja because he's one of the most successful and most recognised Serbian people in the world.
"He accepted that, and he said he would be here in these days.
"I was very much delighted to see him in our box today and watching the match and supporting me.
"It's the first time he actually came to watch me. I've watched him before in the national team a few times. We met briefly.
"He plays for one of the best teams in the world. I was just very happy to see him there in the stands and supporting me."
Djokovic looked in serious danger of losing his first set of the tournament when Berdych gave as good as he got, the pair engaging in some captivating rallies from the baseline.
It only went Djokovic's way when he edged a nervous tie-break, and the world number one then lost two service games in a row to go 3-0 down in the second set. He eventually summoned the mental strength to break back and go on to record victory after two hours and 35 minutes on court.
Djokovic admitted he could have lost the match had he not stepped up his game in the second set.
"It was a very close match, it could have gone either way," Djokovic said.
"He could have won the first two sets, he had a double break in the second.
"I don't know how I managed to go ahead, I don't know how I turned it around. I am really happy with the performance. I am playing some of the best tennis on grass of my career.
"Tomas is a very powerful player. As you could see, we went toe to toe in the first set. It was a huge confidence boost to come back and go two sets up."
Djokovic has won eight of the 11 matches he has played against Del Potro, but the 2009 US Open champion has come out on top in their only meeting on grass, which came in the Olympic bronze medal match last summer.
The Argentinian eighth seed suffered a nasty fall in the opening game of his win over David Ferrer today, but, ever the professional, Djokovic is not taking anything for granted.
"He is a great player. I have a great respect for him," the six-time major winner said of Del Potro.
"He's a grand slam winner. He has struggled with injuries in the last few years, but every time he comes back he comes back very strong because he just has this talent and quality.
"It's the semi-finals, so everything is on the table. We both want to win."