The Silver Arrows chased the Ferraris right to the chequered flag, but it was Sebastian Vettel that took top honours for Ferrari in Sunday's Hungarian Grand Prix.
Silver Arrows end the last race before the summer break in P3 and P4
Vettel led comfortably in the opening stint, but a developing steering issue slowed his charge, and served to back team-mate Kimi Raikkonen into the chasing Mercedes cars.
Valtteri Bottas took the first run at Raikkonen, before the team opted to give Lewis Hamilton - faster at that stage of the race - an opportunity for the final 20 laps.
The pair swapped positions and Lewis made his move, but despite closing to within DRS range, he couldn't pass the Ferrari.
To honour the earlier swap, Lewis allowed Valtteri through on the final lap, securing P3 and P4, ahead of a charging Max Verstappen.
Valtteri: 'Lewis and me are still in the fight.'
"It was really tricky today," said Valtteri. "The pace was not so much a problem. But once you get within 1.5 seconds to the car in front of you it becomes so difficult to close the gap.
"And that track position was a big benefit for Ferrari today. Even though Sebastian was struggling, no-one could pass him.
"We tried as a team to swap positions, but Lewis couldn't get past them either. We didn't gain any points but I'm glad we tried it.
Valtteri: 'Our situations still is not too bad'
"I was promised that Lewis would let me back if it didn't work out. I was struggling with the back markers so the gap to Lewis became bigger than I wanted. But Lewis and the team kept their promise and we swapped positions back in the last lap.
"I don't think every team-mate would do that in a championship fight, so I think that was really nice of him and it shows that he is a real team player.
"Our situations still is not too bad with regards to the points and the championship standing. Lewis and me are still in the fight."
Lewis: 'I tried my best out there'
It was a tough afternoon for Lewis, who spent the first stint bottled up behind Verstappen and lost his radio for a large chunk of the race, but the Brit was still fourth after his last lap swap with Valtteri.
"I tried my best out there," said Lewis. "It's tough when you push so hard and you work so hard and you end up in the same place that you started.
"When the radio didn't work, I was thinking that the team was worried about the tyres not going the distance. So maybe they were going slow so that they could speed up later.
Lewis: 'I know that we can win it'
"I was pushing and I had all this pace, but I was stuck behind Valtteri and I couldn't tell the team because of the radio.
"In the final laps, I kept my word. I said that if I couldn't pass Kimi then I would let Valtteri back. I have said through this year that I want to win this title in the right way - and perhaps I will look at it in a different way if, at the end of the season, I lose out by a small margin - but I believe in doing the right thing and that when you put good things out there, they come back to you.
"I was 20 points behind before, but down to one, and now back to 14. We have lost quite a few points in the first half of the year - but I know that we can win it, once we come back even better after the summer break."
Toto: 'Today was a tough day for the team'
While it was Ferrari, rather than Mercedes that signed off for the summer break with the high of a 1-2, Toto Wolff says the Silver Arrows should be mighty proud - on a day when the team stood by their values.
"Today was a tough day for the team but one that showed the values that we hold high," said Toto.
"First of all, we cannot ultimately be satisfied with third and fourth places; we managed to limit the damage to Ferrari, and were fortunate that Red Bull accounted for themselves on the opening lap.
"So it's positive that we didn't lose too many points on a circuit where Ferrari had the upper hand. But it was also a great day for the team because we saw our ethos played out for everybody to see with great respect between Lewis and Valtteri.
"First, for Valtteri to let his team-mate past to put the pressure on Ferrari and try to challenge Kimi; second, for Lewis to sportingly give the place back in the final corner without losing any position to Verstappen.
Toto: 'These are the values that helped us win six championships'
"These are the values that helped us win six championships and, in the long term, it is the approach that will win us many more titles - even though it was a tough call to make and it's not a great feeling afterwards. But we stick to our principles and our word.
"The race was made more challenging by problems in the garage that cost us radio comms with the cars at points during the race and also affected the data we had in the garage.
"That particularly affected our communication with Lewis and perhaps had an impact on the outcome today, as we could have played our strategy differently if the comms had been working properly.
"But now we have the time to take a step back, recharge the batteries and come back with renewed energy in the second part of the season. There is still everything to play for and both championships are wide open."
James: 'It required a great deal of trust'
The team may have missed out on the win, but Silver Arrows Technical Director James Allison believes the trust and respect shown between Valtteri and Lewis demonstrates qualities beyond the worth of any single victory.
"We would have preferred to head into the summer break on the back of a result like the one we enjoyed in Silverstone, but in its own way this was a race that demonstrated many of the truly great things about this team," said James.
"For the drivers to treat each other with such a degree of respect, that they both could have a go at attacking the Ferraris but then reverse the positions under intense pressure, was both extremely challenging and required a great deal of trust.
James: 'To do it in such a professional manner was a great thing'
"To do it in such a professional manner, with the looming threat of Verstappen running right behind Valtteri, was a great thing.
"After a well-earned summer break, we will look forward to resuming the fight in this intense and intriguing championship, and all our focus will be on ensuring that, come the end of the season, the Mercedes flags are flying high."
Lewis trails Vettel by 14 points in the Drivers' Championship, with Valtteri a further 19 points back after Hungary.
The Silver Arrows head into the summer break with a 39 points advantage over the Scuderia in the Constructors' Standings.