Middlesex 142 for 8 (Dawson 6 for 38) trail Hampshire 419 (Dawson 141, Gubbins 120, de Caires 7-144) by 277 runs
Spinning all-rounder Dawson was overlooked by England for the previously retired Moeen Ali when Jack Leach suffered a lower back stress fracture for the first Test against Australia, before teenager Rehan Ahmed jumped ahead of him for the upcoming second Test.
But having been decisive in his 141 with the bat, he tore through Middlesex with his left-arm spin to claim six for 38 - and leave Middlesex 142 for eight at close.
Dawson had shrugged off being slighted by England the previous evening, saying: "It doesn't really affect me. I've not expected to play Test cricket for a long time. To me if it happens then brilliant, if not then it doesn't bother me."
The clamour behind Dawson has been backed by a strong start to the season with both facets of his game and a deep level of experience.
His 16 wickets is the most for any Championship spinner who is English-qualified and fit - only Leach and South African Simon Harmer have more - while his 384 runs is in the top 30 in the top tier. Add to that 13 Vitality Blast wickets and his finisher runs coming at a 176 strike rate.
He was introduced in the 12th over, after Mohammad Abbas had already nicked off Mark Stoneman, and immediately started with a double wicket maiden.
Pieter Malan was hit on the back pad before Max Holden stumbled in front of a turner.
Sam Robson tried to hang around with 38 off 81 but dangled behind to leave Middlesex in a troubling 66 for four.
After a period of regular lbw shouts and tightness from both ends, Dawson's bowling took another aspect; he added exaggerated bounce to his turn.
Ryan Higgins was the first to be surprised when he edged behind, two balls later John Simpson fended to short leg and then Luke Hollman stepped back to be pinned in front of his stumped. For Dawson it was three wickets in six balls.
It also handed him his second sixth wicket haul in successive home matches, and left the visitors 88 for seven.
De Caires and Toby Roland-Jones scored 48 swiftly before the latter swung Felix Organ to Dawson at midwicket. De Caires ended a personally successful day unbeaten on 24, but his side in danger of being asked to follow on.
Dawson had begun his day with bat in hand continuing his mammoth fifth wicket stand with Nick Gubbins, with all eyes on the former Middlesex man's quest to reach three figures.
It eventually came with a steer down the third after 14 more dots to add to his 277-ball vigil - by far the slowest of his 15 first-class centuries.
He eventually fell for a 318-ball 120 when de Caires had him caught at backward point on the reverse sweep, with Dawson caught at slip playing the same shot soon after.
There were scores of 40 for James Fuller and Felix Organ as Hampshire slowly pushed over 400 but other than Hollman having Fuller stumped, it was de Caires' moment.
De Caires has been seen primarily as a batter growing up but the 21-year-old took the lead with his off-spin - with each wicket celebrated with genuine glee.
His 47.3 overs during the innings was more than the 46 overs combined he had previously managed during his embryonic first-class career, for a single wicket. Keith Barker was lbw to one which stayed low, Kyle Abbott was caught swinging to first slip and Organ holed out to deep midwicket.
His seven-for better his father, former England captain, Michael Atherton's leg-spinning best of six for 78 - and was the seventh-best away Championship bowling figures at the Ageas Bowl.