Sainz and Leclerc were running third and fourth after Red Bull's Sergio Perez had passed both drivers and moved into second.
Vasseur opted not to tell the drivers to hold station and bring home the points and instead, the message was passed on that they could continue to race each other to the flag, but with the proviso of "no risk".
Leclerc made several bold attempts to pass his team-mate, even locking up at the first corner at the start of the final lap, but in the end, Sainz stayed ahead to claim the podium spot.
"If I froze the position you will have exactly the same question – why did you freeze the situation?" said Vasseur when asked about the decision.
"For sure it's much easier to comment when you have it at the end, but I think it was also the best way to thank everybody, the support of the tifosi and so on, and I was not that comfortable to freeze something five laps before the end.
"I trust them, but I told them no risk at all. It's always relative, and I think that the notion of no risk is relative. But it's an opinion, and I'm quite proud of the decision, and of the job done by the drivers today."
Vasseur insisted that as team principal he personally made the decision to allow the drivers to race.
"On this, I want to have the last call," he said. "And I told them no risk, you can race but no risk. Again, it's relative. But I was much more comfortable with this situation than to freeze something."
He added: "You can always say that it was a bit too much or not. But in the end, it was why we finished the race, and I'm more than happy with the outcome of this."
Despite being the driver having to soak up the pressure in the closing stages, Sainz insisted that it was a "clean" battle with his team-mate.
"We obviously knew that the car that is in DRS is always going to be feeling like it's the quicker car," said the Spaniard.
"But at the same time, we know we were both fighting for a podium in Monza. So, there was always going to be a bit of a fight, a bit of a battle. In the end, we kept it clean. There were a couple of nice moves here and there, tight battles.
"But I think today, honestly, I enjoyed battling Max [Verstappen], battling Checo [Perez], battling Charles. I think it was a good day for F1, a good show. And I just did everything I could to stay in front and it worked."
Asked if he'd had any concerns, he said: "It never felt too much of a risk. Obviously, with a team-mate, you're always a bit more tense, [and give] a bit more space because it's the last thing you want in Monza in front of the tifosi, the two Ferraris to touch.
"But I think we had a tough battle. It was for an important position, and we managed to keep it clean."
Leclerc echoes Sainz's sentiments, stressing that he had enjoyed the fight
"I was on the limit, Carlos was on the limit," said the Monegasque driver. "This is normal. I think it means to both of us a lot to be on the podium in front of the tifosi, so we gave it all, but we also know how important it is for the tifosi to have one red car, whoever it is on the podium. So we had that in mind. But we also really enjoyed it. It was cool."
On the order not to take risks, he added: "We both did, as I said. I mean, Carlos was on the limits of the regulations on braking, and I was on the limit of regulations attacking, so we both did, but it all ended well, so it's all fine."
