At the season-opener in Bahrain, Alonso qualified sixth but slipped to ninth behind the Mercedes and McLaren drivers in the race, finishing 74 seconds adrift of winner Max Verstappen.
Alonso qualified fourth for the Saudi Arabian GP, but he soon lost a place to Oscar Piastri's McLaren.
He then remained in fifth place for the duration of the race, successfully staying in front of the Mercedes of George Russell and finishing 35 seconds behind Verstappen.
After a disappointing race in Bahrain, Alonso agreed that Jeddah represented a step forward.
"I think it was better," he said. "We did introduce a new part on Friday that it worked really well. And I think we made a step forward in terms of performance.
"But we still miss maybe two or three tenths compared to McLaren and Mercedes, and we will keep chasing that kind of performance. It is challenging, but it's a nice challenge and an interesting season ahead."
He added: "Happy with the race result, in front of both Mercedes, one McLaren and one Ferrari. It's the maximum I think we can wish at the moment.
"We saw again in the race that we still miss two or three tenths compared to Mercedes and McLaren, maybe a little bit more compared to Red Bull and Ferrari. While in qualifying we seem pretty close. So yeah, we need to keep working on the race pace."
Alonso admitted that he hadn't expected to be able to stay ahead of Russell, believing that the W15 had better pace.
"It was stressful because I was pushing and Oscar was just pulling away," he said. "And George was significantly faster behind, so I was thinking, okay, it's 43 laps to the end after the safety car.
"In FP2 we normally do nine or ten laps on the long run. So to do 42 was, let's say, an unknown territory in terms of tyre age. So yeah, I was just making sure that the strategy was still a one-stop, and we didn't change the plan."
Aston Martin team principal Mike Krack said that it was too early to have a clear picture of where the Silverstone team stands in the pecking order.
"I think we can safely say that we had a better pace," he said. "But also we said after Bahrain let's wait a couple of races to see where we really are, because but Bahrain is really an outlier.
"We have seen that also in the past. So I think when we come back from Melbourne, we have a good picture of where everybody is in terms of first lap, and also in the long runs."
Asked if the Jeddah performance was circuit-specific or related to changes, he said: "I think both. Our Friday race runs were not fantastic.
"And then we tried to change the car a little bit, to help the tyres, to protect the tyres. And fortunately, we did not lose anything [in qualifying].
"And then obviously, the track with each lap is improving, improving. When you have such high grip and high lateral corners, then the rubber that goes off the tyres goes onto the track, and then it gets easier."
