Alonso, a two-time Formula 1 champion, finished Sunday's race in 11th place, coming in 15 minutes behind Zala, reports Efe news.
The Lithuanian racer, who was at the wheel of a Mini 4x4, completed the timed 319-kilometre (198-mile) segment of the first stage of the Dakar Rally between the cities of Jeddah and Al Wajh in 3:19:04.
French Mini driver Stephane Peterhansel, a three-time Dakar Rally champion, finished in second place, just two minutes and 14 seconds behind the winner, while another Mini driver, Carlos Sainz, took third place, finishing two minutes and 50 seconds behind Zala.
Qatari Toyota driver Nasser Al Attiyah, the defending Dakar champion in the car category, posted the best partial times but had tire problems in the final stretch and was unable to win the stage.
Spain's Joan "Nani" Roma, who finished in second place in the 2019 general standings, had a rough debut with Germany's Borgward Rally Team, finishing 28 minutes behind Zala.
The Lithuanian racer took a stage that rally fans had been looking forward to because of Alonso's Dakar debut.
The 38-year-old Alonso managed to put in a solid performance, just missing a top-10 finish.
The second half of the rally, which will start after a rest day in Riyadh, should feature a smaller field, Alonso said.
"That's the idea, to get through the early stages without overdoing it or doing anything extra, and to learn from the race while we're in it," the Spanish driver said.
In Saudi Arabia, the Dakar Rally, which ends on January 17, has a route covering nearly 8,000 kilometres (4,970 miles), with more than 5,000 kilometres (3,106 miles) timed. The race started in Jeddah and will finish in Qiddiyah, near Riyadh.
( With inputs from IANS )