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The Welsh swimmer achieved the qualifying standard for the 400 metres freestyle with a time of four minutes 4.33 seconds in Glasgow.
Carlin had won the 200m and 800m earlier in the week, but her final swim proved decisive.
"I did it on the last swim, so I left it a bit late but I couldn't be any happier right now," she said.
Carlin secured her berth in Rio along with Siobhan-Marie O'Connor and Ben Proud on the third day of the British Championships.
The 25-year-old, who failed to qualify for London 2012 after illness interrupted her preparations, said: "I was trying not to focus on the time too much.
"I was trying to relax and enjoy the race, enjoy the atmosphere, enjoy the Olympic trials, whereas towards the start of the week I was stressing myself out, being so tense and really moody with everyone around me."
Carlin acknowledged her 2012 heartbreak had pushed her on.
"I watched the Olympics. I had a month or so out of the water and I thought 'you know, I can give this one more shot'," she said.
"It was amazing to watch the GB swimmers race but I don't want to watch on the sidelines - I want to be there as part of the Olympics.
"There's a saying that failure pushes you to success and I've really used that, the real lows, to bring me up to where I am now.
"This week has been tough. I have really put pressure on myself, so it feels like a massive weight has been lifted off my shoulders."
Jazz Carlin experienced huge relief and joy after realising her Olympic dream with her third British title in a week.