Having beaten Fabio Capello's Russia last month, Michael O'Neill's side looked on the verge of an even more remarkable result over opponents ranked seventh in the world by FIFA.
With an hour played Northern Ireland were 2-1 ahead, goals from the outstanding Gareth McAuley and Jamie Ward putting them in front after Bruno Alves' opener.
Portugal also had a numerical disadvantage after Helder Postiga's sending off late in the first half but that was levelled up when Chris Brunt picked up a second caution.
The visitors finished the game with a man more when Kyle Lafferty was sent off for a needless lunge, while Cristiano Ronaldo's 15-minute hat-trick - all scored after Brunt's departure - ensured a 4-2 win.
"I think we missed a great chance to get another big scalp," said Davis.
"It was nicely set up for us. The fans were right behind us again and it felt like it might be another one of those special nights, but when we had Chris and Kyle sent off it made it very difficult.
"I didn't really feel like we were going to lose the game, but once we were down to 10 men - and then nine - it was extremely difficult for us.
"They have a lot of quality and it shone through in the end for them."
McAuley, who was an inspirational figure for his side, tweeted: "Thanks for all your support. They were on the ropes. One that got away."
Brunt, meanwhile, took to the same platform after the match to hold his hands up.
"Sorry to all the NI fans tonight," he wrote.
"I should have known better than that and I can only apologise for letting them back into the game.
"The support was brilliant and I can't express how sorry I am for letting everyone down."