The pair put on 181 from less than 40 overs to give the hosts a chance of what would be a remarkable victory, after Gary Ballance's superb ton earlier in the day left Rogers's men chasing 472 for the win.
The Australia opener and his compatriot scored rapidly in the chase though, helping themselves to boundaries at ease as they progressed at more than four runs per-over.
Captain Rogers was in particularly fluent form, punching brilliantly square of the wicket and reaching his 67th first-class century in just 110 balls, featuring 19 fours.
Robson, whose chances of a place in the England side to face Sri Lanka at Lord's next month seem to increase with each match he plays, looked set for his second century of the summer, before he fell to Jack Brooks for 77 after his innings stagnated slightly.
But Dawid Malan (28) put on an unbeaten 49 with Rogers as they successfully negotiated their way to the close on 230/1.
Day three seemed a million miles away from the first half of the match, where wickets tumbled and batting was difficult.
Robson's fellow England prospect Ballance played a quite sublime innings, after resuming on ten overnight, and there were crisp cameos from Adil Rashid (43) and Andy Hodd (22).
Ballance saved his most entertaining batting for his final-wicket partnership with Brooks, smashing his way to a century and plundering 60 in just five overs. His eventual score of 130 in 143 balls featured six sixes.
That was the most fun Yorkshire were to have on a remarkable third day though, as Rogers and Robson wiped the smiles from the faces of the visitors by the close.