The Crescent-Stars entered the contest at the top of Group F, having recorded a 3-1 victory over Georgia in their opening match of the summer competition.
However, Turkey found it difficult against Portugal, with Roberto Martinez's side recording a 3-0 win to secure their spot in the round of 16 as section winners.
Turkey now need to avoid defeat against Czech Republic in their final match in the group on Wednesday evening in order to advance to the knockout round of the competition.
Montella's side fell 1-0 behind in the 21st minute of Saturday's clash, with Bernardo Silva finding the back of the net with an emphatic finish.
Mistakes cost Turkey against Portugal
Portugal then doubled their lead in the 28th minute following a mix-up between Turkey goalkeeper Altay Bayindir and defender Samet Akaydin, with the latter scoring an own goal.
There appeared to be no danger when Joao Cancelo had failed to find Cristiano Ronaldo with a pass, but Bayindir came out of his goal, with Akaydin then over-hitting a pass towards the Manchester United stopper, and the ball rolled into the back of the net.
Turkey then conceded a third in the 56th minute, with Ronaldo setting up Bruno Fernandes to roll into the back of the net from close range.
Montella has insisted that his side competed with Portugal for long stages of the contest but were ultimately undone by defensive mistakes, with the third goal of the match also coming after Abdulkerim Bardakci had failed to make a clearance.
Turkey must now avoid defeat against Czech Republic
"We had our chances and we didn't capitalise, they punished us for every little error. It was an even match and the bad luck of the second goal - which can happen in football - changed the balance of the game," Montella told reporters.
"It was a fairly obvious misunderstanding. Every now and again it happens and unfortunately it happened to the team I manage, I would have preferred it to happen to the opposition but there you go.
"For long periods of the match we played on the same level as one of the best teams at this tournament. They were just more clinical than we were.
"They have more players used to playing in matches like this and maybe that made a difference."
Turkey have been eliminated in the group stage of the last two European Championships, with their best-ever display in the competition coming in Euro 2008, when they reached the semi-finals.