The 22-year-old batted to within nine overs of the close for 157 as they progressed to 470-8, a lead of 64.
He shared stands of 121 with George Bailey (62) and 172 with James Franklin, the latter a Middlesex sixth-wicket record against Yorkshire.
Franklin fell for 99 just before stumps as paceman Jack Brooks claimed 5-89.
England batsman Nick Compton's recent decision to take a break from cricket led to Eskinazi's elevation to the Middlesex first team and he responded with 106 against Lancashire last week.
His latest innings was even better as he advanced his score steadily from his overnight 19 not out and went to three figures off 224 deliveries with a boundary off Steven Patterson, having had a let-off on 84 when Kane Williamson floored a slip chance.
By then he had lost Bailey, who was dropped on 36, and Simpson, both to catches by Tim Bresnan, but Franklin settled in well after taking 30 balls to score his first run.
His partnership with Franklin beat the previous sixth-wicket record for Yorkshire of 166 by Norman Featherstone and Larry Gomes, also at Scarborough, in 1975.
Eskinazi was eventually caught at deep mid-wicket off Brooks, having struck 23 fours in all, and Paul Stirling, Ollie Rayner and Franklin followed as Yorkshire hit back late on, but a draw looks by far the most likely result on the final day.

The man tried to drive through Walcot Lane ford, in Pershore, on Saturday morning, the ambulance service said.
"The elderly man driving had managed to get himself out of the vehicle and was treated at the scene by medics for being cold and wet," a spokesman said.
He said the man was "extremely fortunate" he escaped quickly and urged other drivers to avoid flooded roads.
Fire crews helped rescue the man and his vehicle.
Several flood alerts are in place in Worcestershire. But previous flood warnings, which urge for immediate action as flooding is expected, have been lifted.

But on closer inspection, it turns out the Barcelona striker has not swapped La Liga for the League of Ireland.
It is in fact 16-year-old Barry Cotter, who bears an uncanny resemblance to the Brazil World Cup star.
While Neymar will line-up in the Champions League, former Avenue United and Ennis Town forward Cotter will play in the under-17 National League after signing for Limerick.
But he shares his doppelganger's confidence telling the club website: "I'm looking to score at least one or two goals in each game for the rest of the season."
Stevie Eskinazi hit a second successive Championship hundred as Middlesex fought back strongly on day three against Yorkshire at Scarborough.
[NEXT_CONCEPT]
A driver was taken to hospital after his car became "completely submerged" in a ford.
[NEXT_CONCEPT]
On first glance it looks like Limerick have pulled off the transfer of the century and signed Barcelona striker Neymar.