The 10-man Team soldiered on, playing the final 15 minutes of normal time plus the whole of extra-time a man down, but two excellent finishes from Barnes took them through.
Jubilant Team player-manager David Gordon told JC Sport: "What an absolutely superb performance by the boys; I said I wanted us to have a testing match and we certainly got that.
"Playing against a team in the division above with 10 men for 45 minutes, we ground out that win through sheer determination and one of the most gutsy performance I have seen at this level.
"It’s hard for me to put into words how proud I am of this team; more performances like that one and we will go far in this competition.
"The only worry is we are now starting to pick up injuries but we have a big enough squad to cover any absences."