Jones had won the first set 6-4 against Linette, ranked 82nd, when the rain put a halt to proceedings on Tuesday night.
When the pair returned on Wednesday, it was Linette who appeared fresher 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 told Sky Sports afterwards.
“I thought I’d maybe handle it better than the score reflects.”
Jones rallied back in the decider, claiming the decisive break at 2-1 before completing the victory – just her second at Grand Slam level.
She will play American 14th seed Iva Jovic in the second round on Thursday.
“[I’m] really grateful that I just managed to pull through,” said 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 remarkable five-set comeback against Spain’s Martin Landaluce in the first round, but the British lucky loser’s fortune ran out in a 6-3 6-3 3-6 6-3 defeat by Etcheverry.
The Argentine 27th seed recovered from losing the third set and falling a break behind in the fourth, while Fearnley’s 57 unforced errors and 11 double faults ultimately proved costly.
Samuel, 23, won through qualifying to reach round two on his US Open debut, but was beaten 6-4 6-2 6-4 by 18th seed Lehecka.
Meanwhile, British number 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 also been pushed back to Wednesday because of the rain.
