The 21-year-old played 15 minutes as a second-half substitute in a 2-1 defeat against Belarus on Tuesday.
Oxford-born O'Dowda was left out of Martin O'Neill's final 23-man squad for Euro 2016 but is continuing to train with them before they head to France.
"To be a part of the squad has been incredible," he told BBC Radio Oxford.
O'Dowda's campaign in League Two as part of his club's promotion triumph saw him score 10 goals from midfield and make an initial impression with the Republic's Under-21 squad.
His performances for club and country led to a surprise inclusion in the senior preliminary squad.
But he could not prevent his country from suffering defeat in their final warm-up match - a performance assistant manager Roy Keane called "a reality check".
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"It was a big moment for me," O'Dowda added. "Getting promotion from League Two and then making my senior international debut this season, it's a dream come true really.
"Obviously you see those people on television and I've been asking for loads of advice.
"I think I did alright on my performance. I obviously only had 15 minutes, but they (Martin O'Neill and Roy Keane) said to me before I went on just to do what I normally do.
"Get the ball, play with no fear and be comfortable with it - very similar to what I do at Oxford really."
Meanwhile, O'Dowda says he is remaining focused on his future with Oxford despite speculation linking him with a potential move.
"At the moment, I've still got two years left on my contract," he said.
"I've got my agent and my dad who are looking after me, so I don't really want to know what's going on.
"It (the speculation) is obviously around you, on your phone and people ask the question, but I just let them deal with it."
Oxford United winger Callum O'Dowda says his senior international debut for the Republic of Ireland has capped off "an incredible season".