The young Gers have now beaten Berwick Rangers, Stranraer and Ballymena United to set up a fourth round tie against Solihull Moors.
Mayo said: “I think we are making a big statement here about the quality we have got and we have just got to see how far we can take it.
Rangers booked their place in the last 16 of the competition with a 1-0 win over Ballymena United.
The 19-year-old added: “There was a lot going on in that game I think first things first we are delighted to get the win.
“I still think we didn’t really control the game as much as we would have liked. The message was first of all get a hold of the game and you also need to believe in yourself because I think we were kind of playing with the handbrake on in the first-half and we had to really go for it in the second half and that’s when we started to open them up and we got the chance to go and score.
“Part of the learning experience is us being able to stand up to fights and scraps and stuff going on off the ball but that is part of the learning and that is what we need to take into first team football if we step up.”