Nayan Doshi's 6-111 could not help Surrey as Craig White led Yorkshire to a 470-run lead on day three.
Mark Ramprakash (115) completed a fine century early in the day, but Matthew Hoggard and Darren Gough ran through the tail as Surrey fell to 344 all out.
With a first-innings lead of 250 behind them, White set about the Surrey attack and fired seven sixes in his 117 before they declared on 266-7 - 516 in front.
Adil Rashid then removed Jon Batty (14) as Surrey went to stumps on 47-1.
Rashid, who took four wickets in 15 balls on the second evening, again proved the thorn in Surrey's side to curtail any thoughts the hosts had of surpassing the record 410 they scored in the fourth innings to beat Kent in 2002.
It provided further weight to the argument that new Yorkshire skipper Gough was right not to enforce the follow on.
Despite a relatively pedestrian start to their second innings, in which it appeared Yorkshire might let the initiative slip, White soon upped the pace perfectly - striking seven maximums, mainly off the bowling of Doshi over midwicket.
And Gough finally brought the close to the innings, giving the visitors 13 overs at the end of the day to make a breakthrough - with Rashid duly obliging.