Everton lost the tie 2-1 after falling behind in the first half to arguably their only real error before the interval and then conceded a second as they were chasing the game in stoppage time.
Dominic Calvert-Lewin scored an even later consolation goal but there wasn't enough time left to stage a miraculous comeback.
Nevertheless, Unsworth, taking charge of his first game as caretaker boss following the sacking of Ronald Koeman, said he couldn't fault his players and he reserved special praise for 19-year-old debutant Beni Baningime who was put into the starting XI after Morgan Schneiderlin fell ill.
“We stuck to the gameplan and I'm very proud of them," the 44-year-old said. "Nobody likes losing, but there's so much I saw tonight to work with. A great, committed performance.
“I'm pleased, but not too pleased because we lost the game. It reflects well on the players, they have put in that committed display. I'm sure every Evertonian who watched the game was proud of that performance.
“We can't be where we are and expect to win every game from there. We have to start with a performance and I thought we started with a great performance tonight. We have to get on a winning streak very, very quickly, hopefully starting at Leicester on Sunday.
“When I see and hear experienced players helping the young lads out like they did, there's a lot to work with. I'm quite excited to take that on.”