Jones had received the primary set 6-4 towards Linette, ranked 82nd, when the rain put a halt to proceedings on Tuesday evening.
When the pair returned on Wednesday, it was Linette who appeared more energizing as she raced to a 5-0 lead and wrapped up the second set in 28 minutes.
“I’ve never been quite in this position so I was nervous with how I’d quite feel about coming back out,” Jones informed Sky Sports afterwards.
“I thought I’d maybe handle it better than the score reflects.”
Jones rallied again in the decider, claiming the decisive break at 2-1 earlier than finishing the victory – just her second at Grand Slam degree.
She will play American 14th seed Iva Jovic in the second spherical on Thursday.
“[I’m] really grateful that I just managed to pull through,” mentioned Jones.
“She came out all guns blazing – it was incredible how she was playing until about 4-0 and then it started to equal out. But in that situation, hats off if they can sustain that, I just tried to show my northern grit, stick in there and do what I can.”
Fearnley had produced a outstanding five-set comeback towards Spain’s Martin Landaluce in the primary spherical, however the British fortunate loser’s fortune ran out in a 6-3 6-3 3-6 6-3 defeat by Etcheverry.
The Argentine twenty seventh seed recovered from dropping the third set and falling a break behind in the fourth, whereas Fearnley’s 57 unforced errors and 11 double faults in the end proved pricey.
Samuel, 23, received by way of qualifying to achieve spherical two on his US Open debut, however was crushed 6-4 6-2 6-4 by 18th seed Lehecka.
Meanwhile, British quantity three Jan Choinski fell to a 6-3 6-1 7-6 (7-5) defeat by Dutchman Botic van de Zandschulp in a first-round match that had additionally been pushed again to Wednesday due to the rain.