The pair compiled a third-wicket partnership of 269 on Tuesday as they reached 302 for two.
However, Rogers' dismissal for 132 -- one of six victims for England bowler Jake Ball -- triggered a batting collapse.
Hildreth followed six runs later for 135, and they finished day one at Taunton on 322 for nine, with little opportunity remaining to secure the maximum batting bonus points they will probably need to overhaul the top two.
Middlesex, who could take the title with only a draw if Somerset fail to win, plodded to 208 for five against Yorkshire, thanks largely to an unbeaten century from opening batsman Nick Gubbins.
In a day cut short by bad light, Jack Brooks took three Middlesex wickets to keep alive Yorkshire's hopes of a third successive title.
In one of the tightest finishes for years, Jason Gillespie's side, who started the day a point ahead of Somerset, need not only to beat Middlesex but at least match Somerset's points haul.
At the other end of the table, Hampshire, needing a win to stand any chance of escaping relegation for a second successive season, had an excellent first day at Durham.
Captain James Vince and opener Will Smith were both out in the 90s, and Jimmy Adams and Sean Ervine also hit half-centuries as Hampshire closed on 370 for six, earning them four batting bonus points.
If they win, Warwickshire or Lancashire, who are playing at Edgbaston, could go down with Notts. The home side were bowled out for 219, and Lancashire finished day one on 14 without loss.
(Reporting by Neville Dalton; Editing by Toby Davis)
