The 18-year-old, who was handed a shock Scottish Cup start on Saturday after first-choice left-back Ross Forsyth was injured, was invited for a two-week stint with the English club in the summer.
The trial ultimately ended in disappointment due to an untimely injury - but the former Scottish schoolboy international has put all thoughts of what might have been behind him and insisted he was delighted to make a dream debut for the Cappielow club.
Speaking exclusively to the Tele, O'Ware said: "I was down at Middlesbrough on trial in the summer.
"I originally went down for one week, and I did really well so they asked me to go back down. But then in the second week I got injured on the second day, but I couldn't not train so I had to battle through the week.
"I've only just signed a one-year youth contract with Morton from my boys' club, Bonnyton Thistle.
"I was told on Sunday that I'd be in training with the first team all week, and although there were hints, I wasn't officially told I would start until Friday.
"I was nervous before the game and for a while during the warm-up, but after that I was fine and just enjoyed getting out on the pitch.
"It wasn't as hard as I thought it would be, but a lot of that was to do with Grant Evans and everyone around me talking me through the game.
"So it was a good experience and enjoyable more than anything else.
"We played well, and although I thought their goal was partly my fault as I should've maybe just cleared it, we got through to the next round and that's the most important thing.
"I couldn't have asked for a better debut. I think I've come in and done a job, so hopefully next week I'll get another chance."
Before his whirlwind call-up to the top-team squad, O'Ware had featured in just three games for Mick Rankin's Ton under-19 side.
And the teenager's debut display is all the more impressive when it is considered that he is right footed and usually plays as a centre-back or a defensive midfielder.
But O'Ware insists he is capable of playing in a variety of positions and on either flank due to his grounding as a youth-team player with Rangers.
He added: "I was with Rangers for five years, then with Kilmarnock for two before spending the last year with Bonnyton.
"I usually play centre-back or just in front of the defence, but I haven't played there since I came to Morton.
"I've only played three games for the 19s, and all of those were at left-back.
"I'm right footed but I've always practised with my left, at Rangers especially, as they always drummed it into you to use both feet.
"So I never try to avoid using my left foot, and being right-footed means I have the option of going inside as well, so I'm comfortable playing at left-back.
"Hopefully next week I'll get another chance to play there."