Neath's 24-year-old Jones had to win two matches just to qualify for this tournament but is now one of just eight players still standing, with the famous world trophy and £250,000 top prize at stake.
The two qualifiers, both looking to book their first appearance in the last eight, began the final nine frames with Jones leading 10-6. However, Higginson came out fighting and took the first four frames of Monday evening's session with top breaks of 54 and 53.
The mid-session interval stemmed the Englishman's flow and when the players returned to the table, Jones wasted no time in regaining the lead with a run of 50 and quickly closed in on victory with a tremendous 135 break to make the score 12-10.
The Welshman then dominated the 23rd frame to set up a quarter-final with Ali Carter.
"I couldn't get going in the first four frames," reflected Jones after his victory. "But at 10-10 I just thought to myself in my dressing room 'it is best of five now', and I did well to get what had happened out of my mind.
"It was a bit of a race at the end but I managed to get in with a couple of good breaks. The 135 break was timely. I was struggling to get going but that sort of break really kicked me into gear.
"With qualifying it is four long matches I have played now but I still feel really fresh and I am looking forward to my next game. I have played against Carter once before and I won, so hopefully that's a good omen!
"I love playing at the Crucible. I have surprised myself really, I came up here playing pretty well but I didn't think I would handle the Crucible as well as I have done. To win two games here on your debut is pretty good I think."
Ball by ball scoring and results from tournaments