The Swans travelled to Naples having held the Serie A side to a goalless stalemate at the Liberty Stadium before their best-ever continental run came to a crashing halt.
Napoli took the lead at Stadio San Paolo through Lorenzo Insigne before Jonathan de Guzman drew the Swans level before half-time, giving Monk's side a crucial away goal.
Rafael Benitez's side asserted their authority on the tie following the restart however as Gonzalo Higuain and Gokhan Inler sealed victory for Napoli; yet Monk insists Swansea depart with their heads held high.
"I'm disappointed but I'm not going to put that on the boys," he said. "I could not have asked for any more from my players.
"We had more than enough chances in the 180 minutes but my players should be proud of themselves.
"I thought we dealt very well with the atmosphere. They can really hold their heads high and be proud of what we've done.
"I'm gutted for the players, I feel for them. Everything we asked of them, they did it."
Monk believes Swansea's inability to score on home soil a week previous cost his side dearly, but takes heart from the way in which they tested the third-placed Serie A side.
"We knew that we should have come here with the lead," he added. "But nevertheless, we performed well; we had our chances to score goals.
"I think we competed against them for 180 minutes.
"We pushed them until the last minute and that's when you realise what a great team we are."
Monk hailed the performance of Napoli goalkeeper Pepe Reina, who made a stunning save from Wilfried Bony's header to prevent Swansea taking a 2-1 lead midway through the second half.
"The keeper was great," he said. "I thought the quality of their goalkeeping in the two legs was what made the difference.
"But Napoli have a great team, they have good players. We give all credit to Napoli.
"They have a great coach and we wish them all the best for the future."
Benitez felt his side were worthy winners despite being put to the sword by the 12th-placed Premier League outfit.
"We have scored three goals today against a team that has done really well," he said. "We deserved to go through.
"We have to give them credit, they made life difficult for us. They defended well and made us work very hard.
"But we have to be satisfied to have won against a very difficult rival and we are happy to progress."