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.
Stevie Eskinazi hit a second successive Championship hundred as Middlesex fought back strongly on day three against Yorkshire at Scarborough.