The team BikeExchange-Jayco star took the race into his own hands with an attack 40 kilometres from the finish of the 192km route before overcoming Alberto-Bettiol on the steep final climb to power to victory .
Matthews incredible ride was the second stage win for the Australian team after Dylan Groenewegen sprinted to victory on stage 3.
For the 31-year-old Canberran, the win was just reward after two dramatic second place finishes when he fell short to Wout van Aert on Stage 8 and Tadej Pogacar on stage 6.
An emotional Matthews dedicated the win to his daughter and said it was the thought of her and his wife that allowed him to will himself to victory.
"I wanted to show everyone that I am not just a sprinter, I can also ride like I rode today," Matthews said.
"I was just thinking of my daughter on that final climb the whole way up to the finish, and my wife, for how much sacrifice they make for me to make my dreams come true.
"Hopefully today I showed them the reason why we sacrifice so much."
Matthews, who won the 2017 tour de France green jersey, went on a solo attack from the breakaway with under 50 kilometres remaining of the stage before he was caught by Felix Grosschartner (BORA-hansgrohe), Luis Leon Sanchez (Cofidis) and Andreas Kron (Lotto Soudal).
He was able to shake them off but was then caught and passed by Bettiol on the steep final climb to Mende as Matthews struggled to match it with the Italian.
However, Matthews found an incredible second wind to reclaim the lead on the climb before coasting to victory on the short descent to the Mende runway.
Matthews said he was determined to make up for his struggles on stage 13 with a huge performance today.
"Yesterday was a really good stage for me, but it went so bad for me - the team rode in the final to bring back the sprint for Dylan (Groenewegen) and we were too late," he said.
"Today, I just knew it would probably be my last chance. The stage into Lausanne was a good opportunity and I came up second. Then the other stage when I was second again to Tadej."
In the general classification, Tadej Pogacar (UAE Team Emirates) failed to gain anytime on yellow jersey holder Jonas Vingegaard (Jumbo-Visma) despite attacking early on the 192km stage from Saint Etienne to Mende.
The young Slovenian smashed the race apart on the first climb of the day but his efforts were nullified by Vingegaard and his Jumbo-Visma teammates.
Pogacar attempted a prolonged push on the last climb of the day but he could not shake a determined Vingegaard, though all the rest of the general classification candidates dropped away from the front pair.
Geraint Thomas (INEOS Grenadiers) was the last of the GC contenders to lose touch with the young superstars falling a further 17 seconds behind them in the overall standing, yet he still managed to gain time on fourth-placed Romain Bardet (Team DSM).
The Tour de France continues with Stage 15, a stage for the sprinters as the race travels 202km from Rodez to the historic city of Carcasonne. Watch on the SBS SKODA Tour Tracker from 8.55 pm AEST with the SBS and SBS On Demand broadcast starting from 9.30pm AEST.
