Sainz and team-mate Charles Leclerc were forced to lift and coast early in the 24-lap sprint because Ferrari had gone too aggressive on cooling, which left Sainz exposed to Ricciardo in the fight for eighth.
After staring at Sainz' rear wing for over 10 laps, Ricciardo overtook the Ferrari twice into Turn 1 with the use of DRS, but the Spaniard then retaliated on the back straight by getting a better exit down the Senna Esses and picking up DRS himself.
"It certainly wasn't a boring race, so part of my fun box is fulfilled, but my frustration box is also very full," Ricciardo said about his early race fight with Sainz.
"Obviously had pace for points and every time I would pass Carlos, you then give them the DRS for Turn 2. I passed him, I felt like I got it done and then I just wave him by with DRS open.
"It's always been there but that detection seems kind of frustrating, so it kind of encourages you not to pass in [Turn] 1.
"I'll see if strategically I can make better moves tomorrow. But I felt like if I just sat behind him [in Turn 1] I was never going to pass, so I had to make it work."
To make matters worse his second botched overtake left him vulnerable against McLaren's Oscar Piastri, who nicked his position into Turn 8.
Ricciardo then decided to tuck in behind Piastri to use the same tactic as Sainz into Turn 4, but didn't have the traction to stay close enough through Turn 3's left-hand swoop.
He eventually repassed the McLaren with an earlier move on the main straight, but was still frustrated with ninth, the first position outside the points in a sprint.
"There was one lap where I tried to cut back and then have the line for Turn 2 to get a better exit, but it didn't work. I simply didn't have the traction," he explained.
"Leaving enough room for Oscar to get me on the inside, I'm like, 'Come on, man. Don't let that happen.'
"I got him back and then in some free air with a couple of laps to go we were able to get the pace back, caught Carlos. But over the line, we were one lap too short."
Team-mate Yuki Tsunoda also enjoyed an action-packed race in which he was unable to pass Leclerc, but he did catch Mercedes man Lewis Hamilton, who was dropping down the order on worn tyres.
His sixth place saw AlphaTauri go three points clear of Alfa Romeo in the constructors' championship, and is another step towards catching Williams in seventh.
"The pace we showed was a big surprise for me. I feel like we have more pace compared to yesterday," the Japanese driver said after both he and Ricciardo were eliminated in Q1 on Friday.
After an error-strewn Mexico Grand Prix, Tsunoda displayed much better racecraft, showing much more patience while racing Leclerc and Hamilton than he did with Piastri last week.
"I was more patient than probably previous races, especially when I was behind Charles," he added.
"I didn't over push trying to overtake Lewis as well, that for me was good. Flashback from Mexico, but pretty happy with racecraft today."
Additional reporting by Adam Cooper
