Wimbledon is the only major where the American has yet to go beyond the fourth round and the seventh seed was staring down the barrel of defeat against an opponent who last year became the first lucky loser to reach the women's last 16 in the Open era.
Gauff rattled through the first set on Court One but lost her way in the second, possibly unsettled by a bizarre serve that went sideways into the umpire's chair.
Sierra, now ranked 56 in the world and with a direct entry to the tournament, broke the American to love to go 2-0 up and forged 4-1 ahead before ensuring the match went into a third set.
The 22-year-old Argentine then had a break point on 1-1 but Gauff saved it and dished up a mighty 124 mph serve to regain the advantage and go 2-1 up.
Sierra saved a break point for 2-2, and two more to hold at 3-3, before turning things around and breaking Gauff to go 4-3 up.
Serving to stay in the match at 5-3, Gauff held on and then broke her opponent to regain the momentum and leave Sierra on the back foot and fighting for survival.
The Argentine took a 7-4 lead in the tiebreak but Gauff then won six points in a row, capping her performance with a 117 mph ace on the first of two match points.
"Solana played a great match and, yeah, I think it was just tug of war with us," said Gauff.
"She was hitting some big shots so I felt like a lot of times I was on the defence. I was trying to be aggressive when I could but overall just happy with my serve.
"I thought it held up the whole match and obviously it helped me in the tiebreaker."