Ricciardo finished in seventh place under lights at Sakhir on Sunday, while his teammate Lando Norris crossed the line in fourth.
In an exciting race that was won by Mercedes driver Lewis Hamilton, Ricciardo, who started from P6, was passed by Norris on the opening lap.
The Aussie was also part of an incident with Pierre Gasly early on that left the AlphaTauri driver without his front wing.
But Ricciardo held on to stay in the points, despite a lack of pace relative to Norris in the latter stages of the race.
“First one down! I think overall it’s been a good weekend,” said Ricciardo.
“We can’t complain with that result. I think from the team’s perspective as well it’s good to hit the road with fourth and seventh. I think there’s plenty to learn from.
“I wasn’t that impressed with my race. We couldn’t find the pace, so every time I tried to get a bit more out of it, it wasn’t really there.
“But I think there’s plenty to take from the race, plenty to learn from.
“If what I felt was an OK race ended with a P7, we’re looking alright. We’ll take the positives and learn from the data.
“Overall, a good weekend for the whole team, and a good first race for McLaren.”
Team principal Andreas Seidl said it was a great start to the 2021 F1® season for McLaren, who signed Ricciardo from Renault (now Alpine) to replace Carlos Sainz.
“Today the team executed the race very well, while Lando and Daniel drove strong races to bring us home in P4 and P7,” said Seidl.
“It gives us good points in a constructors’ championship that looks already very, very tight.
“We leave Bahrain committed as ever to working hard, improving our car and ensuring we’re still as competitive at the final race as we were at the first.”