Media playback is not supported on this device
They led Patrick Hausding and Sascha Klein by 4.4 points after five dives but a poor final effort cost them.
The Germans won by just 0.96 points, with Ukraine's Oleksandr Gorshkovozov and Maksym Dolgov third.
Britain's Jack Laugher won silver in the three metre springboard.
Daley told BBC Sport: "I think we have lots of positives to take from that performance.
"Yes, we wanted the gold medal, and yes it was really close, but it was the highest score we've got on our first two dives all year, it was only two points less than our personal best and we missed two dives.
"There's still a lot more room for improvement, but we'll let them have this one and take them at the Olympics."
Media playback is not supported on this device
Lauger had been in gold medal position after the first two rounds of six, only for injury to strike before his third effort and drop him briefly to fifth before a strong late recovery.
Russia's Evgenii Kuznetsov took gold, with Ukraine's Illia Kvasha taking bronze.
Laugher said: "I came out really strong, but unfortunately before my second dive I started cramping really badly in my calves. I got a lot of pain coming through them, so on the third dive it was early hard to press out of the board, which made it quite slow, which finished it short.
"It shook me a little bit, but I managed to bounce back a little bit and finish with a good score with some good bits in there. It's just a disappointment about a mishap that cost me my momentum and flow.
"It was about clawing my way back from there, and there's a lesson - if it feels like it hurts and I can't dive, I can still claw it back to a silver medal. So I'm happy with it, and this silver medal boosts my confidence going towards Rio."
Daley still has his individual 10m platform competition to come on Friday, where he will look to take revenge on Klein and add to the gold medal he won with Grace Reid in the the mixed 3m synchronised event on Wednesday.
Media playback is not supported on this device
Britain's Tom Daley and Dan Goodfellow had to settle for 10 metre synchro diving silver at the European Aquatics Championships after slipping out of gold medal position in the final round.