Defender Dawid Kucharski, midfielder Ian Black and Spanish winger Suso Santana have all moved to Tynecastle.
Dawid Siwak, 30, allegedly offered the men work in the UK and then controlled their finances and lifestyle.
But Dawid Malan (87no) helped guide the hosts to 229-7, before a sporting declaration set up the Essex chase.
Dawid Malan ended unbeaten on 84, with Gareth Berg 30 not out.
Dawid Siwak, 30, offered the four men work in the UK, housed them in crowded flats and retained most of their wages.
But only the inexperienced pair of Dawid Malan and Dan Housego offered any prolonged resistance as the wickets tumbled to a rash of poor shots.
Middlesex chasing 190 to win, employed their one day opening partnership of Neil Compton and Dawid Malan to get them off to a solid start.
With the game looking set for a draw, Udal declared on 414-8 after Dawid Malan had made 88, then succeeded in putting the Welsh under real pressure.
"Pensions linked to CPI will be lower over a period of time - some estimates put the drop as high as 25%", said Dawid Konotey-Ahulu of Mallowstreet.
Croft then took three of the next four wickets, trapping Eoin Morgan lbw (17), having Dawid Malan caught by Herschelle Gibbs (47) and bowling Gareth Berg (18).
Middlesex slipped to 114-4 but Ed Joyce shared an unbroken 39 with Dawid Malan to see them home with an over to spare as Sussex slipped to a seventh defeat.
Batting for the second time in the day, Newman perished quickly in Middlesex's second innings, and Owais Shah and Dawid Malan were dismissed in consecutive Footitt overs just before the close.
Dawid Malan passed 1, 000 first-class runs for the season, only to then move across his crease just before lunch to have his leg stump clipped as he missed an attempted flick off his pads.
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