With Australia leading 1-0, Miller fouled Son Heung-Min in the box in second-half stoppage time which allowed South Korea's Hwang Hee-Chan to equalise from the spot and force extra time.
Miller then fouled Hwang on the edge of the area in extra time, allowing Son to score from a free kick in the 104th minute and complete the comeback.
"I put my arm around him and give him a hug, tell him that these things are lessons in life.
"You learn from these types of things and you move forward," Arnold told reporters when asked how he would deal with Miller's errors.
"He's got to get back to his club and it's a different environment from here, he'll be fine."
Although champions in 2015, Australia were not the out-and-out favourites for the Asian Cup this year and Arnold said it was still a good tournament for a team that is growing as they look to rebuild for the next World Cup.
The squad had only three players from the 2019 Asian Cup - captain and goalkeeper Mathew Ryan, Aziz Behich and Jackson Irvine.
"We've got a very young squad in terms of caps, in terms of appearances. We only had 12 players here from the World Cup (in 2022)," Arnold added.
"We're regenerating the squad after the World Cup. That's why you saw a number of new players on the pitch. After this Asian Cup some older players may retire and move on and we've learned a lot from the new players."
Despite their exit, Football Australia CEO James Johnson said he was filled with an "immense sense of pride" for what the squad had accomplished.
"This team, this great group of players, has every reason to hold their heads high, and I have no doubt they will return stronger," Johnson said.
"As we turn our focus to the upcoming Olympic qualifiers and World Cup qualifiers, the lessons learned and the experiences gained from this tournament will undoubtedly fuel our ambition and preparation."
