The Aston Martin driver lost out to third-starting Lewis Hamilton initially into Turn 1 and was later required to briefly touch the brakes at the first round of pitstops when the Mercedes appeared to be released into his path.
This incident was reviewed but passed without a penalty from the stewards.
Alonso then recovered his original starting position with the aid of DRS to pass his former McLaren team-mate into the final chicane on lap 22 of 70.
Hamilton did give chase to close to within two seconds during the final 10 laps before Alonso dropped his times to secure his place as runner-up behind dominant race winner Max Verstappen.
Given the Mercedes threat, Alonso said there was no "one lap where I could relax a little bit" in what he reckoned was an "amazing battle". The two-time champion added that for the duration of the race, he drove as though it was "qualifying".
He said: "I think we were hoping to challenge a little bit more the Red Bull, but we lost a place at the start with Lewis.
"Then it was a battle with the Mercedes and Lewis was pushing all the race so I didn't have one lap where I could relax a little bit. So, it was an amazing battle."
Alonso reckoned the balance of performance had swung in favour of Hamilton as the race wore on, with the Mercedes W14 taking time to hit its stride compared to the newly upgraded AMR23 that was faster out of the blocks.
He continued: "At the beginning, I had a little bit more pace. At the end, I think Lewis had a little bit more pace. I was tough.
"It was a very demanding race, all 70 laps of qualifying today."
Alonso added that for the next race, the Austrian GP in early July, the team goal would be "putting more pressure on Max" after Aston had instructed him to ease off late on, helping Verstappen to his win by 9.57 seconds.
Alonso said: "I felt the car was OK. But I was just following the instructions. So hopefully, that means that we have a little bit more pace. So next one, putting more pressure on Max."
