Claude Puel's side were the better side for much of an enthralling final, which was settled by a late Zlatan Ibrahimovic header as United recorded a 3-2 win.
The south-coast side fought from two goals down having had an early Manolo Gabbiadini strike ruled out for offside, with the Italian forward scoring either side of half-time to cancel out an Ibrahimovic free-kick and a Jesse Lingard effort.
The outsiders then dominated large swathes of play only to fall to Ibrahimovic's 87th-minute heroics.
While plenty outside of St Mary's have been quick to praise the performance of Southampton, Davis admitted after the game it counts for very little.
Asked about being the better team in the final, the Northern Ireland midfielder said: "It's not consolation at the minute to be honest.
"But if we play like that between now and the end of the season we'll certainly win more games. Everyone's feeling down.
"We can feel proud of the spirit and character we've shown, especially after going 2-0 down. When we got ourselves back in the game, we thought we might go on and win in it.
"We showed great character to get back in the game, but luck wasn't on our side today - the disallowed goal that would have put us in front and then the second half we hit the post and David De Gea had a couple of good saves.
"Some decision you get and some you don't, but in a major final you want those decisions to go for you, especially when you're playing against a team like Manchester United."
Davis, who captained Saints in their attempt to win a first major honour since 1976 - when they beat United in the FA Cup final - said it was United's clinical edge in front of goal that ultimately proved the difference on the day.
"Even when we went behind we thought we were relatively comfortable," he said.
"They took the two chances they got and that's the quality they have in their side. When you get it 2-2 you really start to believe you can win this.
"United were clinical and that's hard. It will take a couple of days to mull it over. It was obviously a quiet dressing room as you'd expect - it's a difficult one to take because we deserved a lot more. You can't really enjoy a final unless you win it."
