Middlesex scored 441 for 5 in their first innings, with standout performances from Higgins who scored 117, Robson who scored 108, Davies who scored 61, and Holden who scored 51. Bess was the pick of the Yorkshire bowlers, taking 4 wickets for 168 runs. In response, Yorkshire declared their first innings at 601 for 6, taking a lead of 160 runs over Middlesex.
Centurion Ryan Higgins became this summer's second batter to reach 1,000 runs in the Vitality County Championship as Middlesex batted themselves to an almost certain draw against Yorkshire on day three at Headingley.
Unbeaten all-rounder Higgins, who followed Glamorgan batter Colin Ingram's lead after he achieved the feat on day two, top-scored with 117 off 210 balls in Middlesex's 441 for five as they replied to Yorkshire's first-innings 601 for six declared.
Higgins posted his fifth hundred of the ongoing Division Two campaign, in his 10th appearance, and was joined in posting a century by opener Sam Robson with 108 off 219 balls.
Higgins was slightly more aggressive against an accurate spin-led Yorkshire attack, as reaching his hundred with a six suggests, while Robson's innings came on the ground where he scored his only Test century for England 10 years ago.
Another ex-England player, Dom Bess, claimed four for 168 in 64 overs of off-spin.
Robson started day three 65 not out, with Middlesex 141 for one, and he went on to record the 35th first-class century of an impressive career including seven Test appearances in 2014.
Having been trapped lbw defending against Bess, before lunch, Robson then watched Higgins build on his good work and see his side to the verge of the 452 follow-on target late in the day.
Robson's England century - 127 - came in the first innings at Headingley against Sri Lanka, a Test the visitors won in June 2014. It was the now 35-year-old's second of seven home appearances before being discarded.
Jonathan Trott acted as a locum opener alongside Sir Alastair Cook for a West Indies tour in early 2015 before Adam Lyth took over for that summer's home series against New Zealand and then Australia.
Ironically, Lyth is playing in this game for Yorkshire and bowled at Robson late on day one.
Lyth also played seven home summer Tests before discarded, and his only century came at Headingley.
Robson was expertly watchful against the rare bursts of seam but mainly spin in the form of Bess, Lyth and Dan Moriarty, and he worked well off his legs.
Having shared 61 for the first wicket with Mark Stoneman either side of tea on day two, he completed a second-wicket 121 with Max Holden this morning.
Left-handed Holden made 51 before feathering behind off Bess, leaving Middlesex at 182 for two in the 64th over.
Robson was trapped lbw playing forwards at Bess with the score on 207 in the 76th, a key time with the new ball around the corner.
Leus du Plooy and Higgins negated that threat, utilised by the spinners, as they added a fourth-wicket 57 either side of lunch, where Middlesex reached at 248 for three.
Du Plooy was ousted early in the afternoon for 33, caught at short cover off a low full toss from new ball seamer Ben Coad, who was bowling only his seventh over of the innings - 263 for four in the 93rd.
While determined Middlesex will be delighted to achieve their mid-match goal of avoiding defeat, this wasn't a day they dominated.
They weren't able to break free and achieve a sizeable haul of batting bonus points as Bess and Moriarty have bowled 116 overs between them in the innings.
Upon reaching 321 for four after 110 overs, they claimed two batting points to Yorkshire's one for bowling. In the match overall, Yorkshire have edged that battle five to three.
Should this contest finish in a draw, as expected, second-placed Middlesex would be one point clear of Yorkshire in third with three matches remaining in the promotion race.
Fifth wicket pair Higgins and wicketkeeper-batter Jack Davies shared 128 either side of tea, the latter adding 61 before edging a low catch to slip off Bess - 391 for five in the 133rd.
Excellent Higgins, who swept and reverse swept well, reached 1,000 Championship runs for the season upon getting to 92 and brought up his latest century with a six over long-on against Moriarty's left-arm spin.
He got there off 185 balls amidst an unbroken sixth-wicket stand of 50 with Luke Hollman, 21 not out.