Allertova, ranked 130th in the world, dominated the first set before staging an impressive comeback from 4-1 down in the second set, winning the last five games to set up a Round of 16 clash with fourth seed Elina Svitolina.
Allertova cruises through first set
It was a fast start for the Czech as she held to 15 and immediately pressured Linette's serve, leading 15-40. Using her flat, hard backhand, she pinned the Pole into the backhand corner until extracting an error off of that wing for a 2-0 lead.
After being tested in the fifth game, being pushed to deuce, Allertova sought about getting an insurance break and after bypassing her first opportunity, a backhand down the line confirmed the second break and a 5-1 lead.
Serving for the set, the qualifier registered an ace, a pair of winners and an unreturnable serve to claim the opener in just 21 minutes.
Czech storms back from early second set deficit to reach first Round of 16 at a major
Riding the wave of a dominant first set, Allertova held a break point in the opening game of the second set only to see Linette fight it off and hold. That seemed to spark the world number 74 as she broke to love, the Czech double-faulting to confirm the Pole's advantage at 2-0.
The longest game of the match ensued, 16 points played and Linette swatting away three more break points for a 3-0 lead. Eventually, she would go ahead 4-1, but after wasting four break chances to take a double break lead, the set and match turned back to the Czech.
Her weight of shot once again having an impact, Allertova broke back when her backhand forced a weak reply from Linette and after an easy hold, she made it four games on the trot, blasting forehands that the helpless Pole could only dump into the net on break point for a 5-4 lead.
With the pressure of trying to serve out the match, the world number 130 built a 40-15 lead and on match point, Linette dumped one final groundstroke into the net, confirming Allertova's passage into the last 16.
Injuries have held Allertova back so far in her career, but when healthy, she is a dangerous player. Generating power with ease, she struck clean winners with ease and showed how strong her serve can be.
After the match, she said "I felt terrible. I didn't know what was happening at the start of the second set. It was tough with the weather. I'm glad I came back. I don't know how to say how I am feeling."
Credit to her for fighting through virtually unplayable conditions and she can now look forward to her first fourth round appearance at a major with little pressure against Svitolina.
Linette's basic problem was that she had no answers for Allertova's power, hitting just nine winners as compared to 23 for the Czech. Although she got 62% of her first serves in, she won only 54% of those points.
With her lack of firepower, the Pole may have wasted her best chance to reach the fourth round of a Slam and although she's made improvements to her game in the last year, she was shown that she still has more work to do to appear at this stage on a regular basis.