Wright took 4/23 in 14 overs - including a spell of four wickets for no runs - as MCC's top order succumbed to a mixture of excellent bowling and loose shots to collapse from 47/1 to 61/7.
The middle order of captain Dale Benkenstein, Dawid Malan and Peter Trego were all dismissed for ducks in a stunning burst from the highly rated Wright, who extracted more from the surface in one spell than MCC had managed in 150.4 overs in the first innings.
Watch: Day two highlights
MCC were not helped by the retirement hurt of England batsman Joe Denly, who was unbeaten on 19 at number three, with a migraine, or the dismissal of opener Sam Northeast, who was run out backing-up at the non-striker's end just before the tea interval to set the ball rolling for Warwickshire.
The county champions had left MCC a tricky 30-minute spell to see out after declaring on a mammoth 561/6 and briefly appeared to find the pitch as comfortable for batting at it had been in the previous five sessions.
But once the Warwickshire had fortuitously got their foot in the door through the combination of Wright's fingertips and a firm Jimmy Adams drive, the match assumed a completely different character.
Wright runs out Sam Northeast
Wright looked in good rhythm from the off and when Denly was forced from the field after tea he came alive, clean-bowling Benkenstein, finding the edges of Adams and Trego's bats and trapping Malan LBW in a remarkable burst which had barely seemed possible earlier, when Warwickshire's batsmen had continued the domination they'd shown on the first day's play.
Resuming on 370/3 they built slowly through Laurie Evans initially before racing to the declaration with some aggressive strokepley from 'keeper Tim Ambrose and the unbeaten Rikki Clarke and Ateeq Javid. Chris Rushworth was the pick of the MCC seamers, finishing with 3/93.
It seems unlikely that Warwickshire will be called upon to bat again, and MCC know they need a remarkable performance - or even the intervention of more uncustomary desert rain, which had the groundstaff reaching for the rarely-used covers at the Sheikh Zayed Stadium during the lunch interval - in order to salvage a draw.