After taking an emotional pole on Saturday, Sainz produced a combative display at Monza in front of Ferrari's tifosi, putting on a robust defence in the first stint of a one-stopper in an attempt to keep Verstappen's quicker Red Bull at bay.
Sainz used his Ferrari's excellent top speed to successfully defend his position several times into the first two chicanes. But following a lock-up into Turn 1 on Lap 15 the Spaniard had to allow Verstappen through into Turn 4, and soon had to focus on safeguarding second place against Verstappen's team-mate Sergio Perez.
Sainz ultimately settled for third behind the two Red Bulls, beating team-mate Charles Leclerc to the coveted final podium spot as Ferrari allowed the pair to continue their spirited battle to the finish.
But afterwards he revealed there was a brief moment in the first stint when he believed he could pull off a shock win in front of the adoring Italian home crowd, until he realised he had pushed his tyres too far in an attempt to keep Verstappen behind.
"Yeah, halfway through it," he said. "I felt like I had it fairly under control but then around lap 10 to 12 I started feeling the rear left tyre giving up a lot, a lot earlier than I would have expected.
"At that point, I realised I had used my tyres too much, probably to keep Max behind.
"I had worn that rear-left tyre too much and I was going to suffer a lot for the rest of the race because it was going to probably make me box early onto the hard tyre, and then the second stint was going to be very long.
"It’s exactly what happened, and my feeling was correct. Honestly, I didn't expect to degrade so much but it was clear that I was pushing very, very hard to keep them behind. Probably harder than what I should have done."
Sainz conceded third was the best possible result as Red Bull had prioritised its race pace over qualifying, and its extra downforce compared to the Ferraris stood it in good stead as it once again enjoyed superior tyre management.
"Obviously I'm very, very happy now, because a P3 in Monza in front of the tifosi is as good as it can get," Sainz added.
"At least for this weekend, because clearly Red Bull was in the end quite a bit quicker than us today, as we expected.
"I tried everything I could to keep them behind, especially that first stint in front of Max.
"It probably nearly cost me a podium, because it meant that I was wearing a lot my tyres and using the tyres more than probably what I would have liked, and making me very vulnerable towards the end of the stints.
"I was always under a lot of pressure by Max first, then Checo, then Charles. In the end I could make it on to P3 because we made it, but it was a tough one. A very tough one."
