Second seed Ruud, the world number seven, is at home in the Swiss clay court tournament, having won the Geneva title in 2021 and 2022.
He beat Italy's in-form Cobolli 1-6, 6-1, 7-6 (7/4) in a roller coaster match on the Parc des Eaux-Vives clay.
The contest betwen two-time French Open runner-up Ruud and Cobolli, who is on a career-high ranking of 56, was meant to take place on Friday evening but was called off due to persistent rain.
"It was very, very difficult. I was one point away from losing," said Ruud.
"I just was able to hang in there and it was a really tough match. I was happy to be able to come back in the third set and pull through.
"It was a good confidence win for me.
"I'll just take a shower, rest, have some lunch and then start again," added the 25-year-old.
World number one Djokovic is the reigning French Open champion and the Serb admitted he was "worried" about his title defence after falling to 44th-ranked Czech Machac in the Geneva semi-finals on Friday.
Djokovic took a late wildcard to play in Geneva in a bid to reverse an alarming dip in form ahead of Roland-Garros, where he has won three times.
But the 24-time Grand Slam champion went down to a 6-4, 0-6, 6-1 loss to Machac, competing in his maiden tour-level semi-final.
The last Czech to reach the Geneva final was Tomas Smid in 1987.
The 28-man Geneva tournament serves as a warm-up before the French Open, the second of the year's four Grand Slams, which begins on Sunday.
