The National League North part-timers defied the odds in reaching the final and were hopeful of pulling off one more upset against Aldershot Town.
But the quality of the National League side told in the second half, with goals from Jack Barham, Dan Ellison and Josh Barrett securing the victory at Wembley.
But Lee says Spennymoor’s players should and will celebrate their run to the final.
“I’ve thanked every one of the players for what they’ve achieved, what they’ve done and their efforts,” said Spennymoor’s boss.
“It’s been an immense challenge this season, the amount of games we’ve played for a part time club, the travelling we’ve done.
“These lads are full time workers and we demand effort like you would a full time player. You can’t be anything but proud of what these lads have achieved.
“The lads have played at Wembley now, you can’t take that away from them.
“Walking up those steps, even though we lost, we have to be immensely proud of what we’ve achieved.
“We’ll have a beer tonight and celebrate what we’ve done this season.
“It’s been an unbelievable run. To even get close to Wembley for a club like us is massive, to achieve it is huge. It just wasn’t to be today but I’ll smile tonight. I’m very one proud man.
“Aldershot will celebrate, rightly so, but we’ll celebrate ourselves.”
Spennymoor were more than a match for Aldershot before the break and had the best chance of the first half through Rob Ramshaw.
Lee said: “That first half chance, it was a fantastic run, pulled across and watching back, I think Rob just loses his footing a little bit and doesn’t connect well.
“A million times out of ten you’d back him to hit the back of the net. Games and finals are about moments. We’ve had moments throughout this cup run. Today we just didn’t take them.
“It’s mixed emotions. I’m extremely proud of what the lads have achieved to get here. We spoke about not making it about just getting here but trying to get a result.
“Games are all about moments and we just didn’t take them today. Even their first goal, we were in a good position on the edge of their box. We lose it, they counter and five second later we’re 1-0 down. That seemed to deflate us and took the energy out of the lads. We just couldn’t get ourselves back into it.”
