The on-loan Juventus midfielder, brought on in the second half of extra time, missed the only spot-kick as Frankfurt prevailed 5-4 in a game that finished level at 1-1 in Seville on Thursday (AEST).
It marked the first time Rangers had tasted defeat on penalties in European competition since the 1999-2000 UEFA Cup when losing to Borussia Dortmund in the last 32.
Ramsey was visibly upset after Rafael Borre converted the decisive penalty for Frankfurt, who are the first German team to win the UEFA Cup or Europa League since Schalke in 1996-97.
But Page insists he has no worries about selecting the 31-year-old to take a penalty should Wales' upcoming World Cup qualifying play-off against either Scotland or Ukraine go the distance.
"I have no doubt whatsoever that if it comes to penalties that he will take one," Page said at a news conference on Thursday after naming his 27-man squad for that June 5 showdown in Cardiff.
Asked if he still has faith in Ramsey, Page said: "Absolutely, 100 per cent. It happens to the best in the world. Top players, they put themselves up.
"I am so proud of him for putting himself up [to take a penalty]. He is a world-class footballer. He had the courage and conviction to take a penalty. Top players recover from that, no problem."
Ramsey is part of Wales' latest squad, as is Gareth Bale, who will become a free agent when his Real Madrid contract expires in the coming weeks.
Bale's agent this week suggested a return to English football is the most likely option, with the forward aiming to keep fit ahead of November's World Cup should Wales qualify for the tournament for the first time in 64 years.
"I think everybody is probably thinking the same thing about his future," Page said. "It will depend on how results go with regards to what he does next season with a club.
"The only person who can answer that is Gareth. All I know is whenever he turns up for us he is first class. The full focus on himself is winning this game against Ukraine or Scotland, so he can worry about that after."
Bale has started just four games for Madrid in all competitions this season and has not been part of their matchday squad for any of their past five games because of a back problem.
The Wales skipper is back in training, however, and Page expects him to be part of Los Blancos' squad for next week's Champions League final against Liverpool in Paris.
"I think Gareth puts himself when he is fit to be available for Real Madrid like he does with us," Page said. "It is down to the manager then whether he wants to select him or not in the squad.
"I think he is confident he will be involved and we will wait and see whether he is involved. We will then get him in camp as soon as we can."