Wild card Murray was 7-5, 4-1 down to Yannick Hanfmann when a thunderstorm lashed the Parc des Eaux-Vives grounds on Monday.
They resumed on Tuesday, with the German world number 85 taking little more than 10 minutes to win the second set 6-2.
Ever since Friday's draw, tennis fans had been relishing what could perhaps have been the last match between members of the Big Four who have dominated men's tennis this century: Djokovic, Murray, Rafael Nadal and the now-retired Roger Federer.
Instead, Hanfmann will now face world number one Djokovic in the second round on Wednesday.
Now number 75 in the world, Murray, aged 37 and playing with a metal hip, has said he is unlikely to carry on playing competitive tennis beyond the next few months, while Nadal, also plagued by injuries, is likewise facing the end of his career and is unsure if he will make a farewell appearance at next week's French Open.
The Geneva match was former world number one and three-time Grand Slam winner Murray's first tour-level outing since damaging tendons in his left ankle in Miami in March.
The 28-man Swiss clay court tournament serves as a final tune-up before the French Open, the second Grand Slam of the year.
Djokovic, who turns 37 on Wednesday, took a wild card to play in Geneva in a bid to rescue an alarming dip in form ahead of his French Open title defence next week.
The record 24-time Grand Slam champion will arrive in Paris without a title in the season for the first time since 2018, unless he takes the Geneva trophy.
Casper Ruud, the Norwegian world number seven, is also in the Geneva draw, along with fellow top 20 players Taylor Fritz, Ben Shelton and Sebastian Baez.
Murray and Djokovic had last played in 2017 in the Doha final, with the Serb taking the title.
Murray won the US Open in 2012 and Wimbledon in 2013 and 2016, making 11 Grand Slam final appearances in all. He also won the 2012 and 2016 Olympic singles titles.
