The world number 11's success came after Dan Evans and Neal Skupski defeated Sebastian Cabal and Robert Farah in the doubles earlier in the day to put the visitors ahead for the first time in the tie.
"It was a good day, with Dan and Neal coming out strong, they came out firing with a tough ask against some of the best doubles guys in the world, especially in these conditions in Bogota," Norrie said.
"It really gave me a lot of confidence going into my match, being 2-1 up rather than 2-1 down. Mejia was tough, he was trying to get the crowd involved and on his side, and I really made the court mine and I was able put enough balls in the court and I served really well again.
"The team was fired up and it wasn't an easy task for us to get through so Leon and the rest of the guys did a really good job."
The day began with the score level at 1-1 after the sides took a victory each in Friday's opening singles rubbers.
Norrie, who beat Nicolas Barrientos on the first day, saw off Mejia in just over two hours on his second match point.
Norrie's win meant Evans, who lost to Mejia on Friday, did not need to play the final match against Barrientos. Britain now go through to the group stage in September.
Captain Leon Smith said he had "immense" pride in his team's performance, which was produced despite the short turnaround following the Australian Open.
"Feelings are one of immense pride for what the boys have done, and the support team - everyone's committed a lot of time to this," Smith said before returning to the UK.
"Hats off to the players - they came down to Colombia, most straight from Melbourne, came earlier than we would ever normally do for a Davis Cup, it was a big commitment from them, but we did that to prepare best for what happened these last two days and it's definitely paid off.
"There's a lot of work that the players have done to adapt in a short space of time and they went out and fought really hard.
"It never had to be the highest quality of tennis because conditions don't allow for that, but it needed a lot of grit and determination.
"The important thing is we got a win, which gives us a chance to keep being successful this year and (we're) really looking forward to September now.
The man who led Britain to victory in the Davis Cup in 2015 added: "I just can't thank the players enough for the commitment they've given - and all the support team, as well, that have done the same."
Davis Cup Finals Group Stage tickets booked for Great Britain â #DavisCup | @the_LTA— Davis Cup (@DavisCup) February 5, 2023