The Democratic Republic of Congo international crossed along the six-yard line to find the unmarked Cole, in his first game for six weeks after a groin strain, who calmly slotted in his 12th of the season.
Boro needed a miracle, in fact several miracles knowing they had an inferior goal difference to Hull, but they were given a glimmer of hope immediately after the break.
Tuncay had the West Ham defenders trailing in his wake after a mazy run down the middle and the ball fell to O'Neil on the right who wasted little time in crashing home his fourth of the season.
Off the bench came Joe Bennett and Josh Walker after the interval and Boro's tails were soon pointing upwards after midfielder Matthew Bates went close with a 30-yard piledriver.
But parity lasted a mere eight minutes as they conceded a very soft goal which sent Southgate's head bowing into his hands.
In a sweeping move on the edge of the box, Jack Collison passed into Stanislas and his low shot should have been dealt with by Brad Jones, but the ball slipped through his fingers.
Victory 'perfect end' to season - Zola
Kieron Dyer was introduced for only his eighth appearance of the season and the former England star was unlucky not to score after a fine turn in the box.
Southgate said before the game he was already preparing for next season in the Championship and when the disappointment recedes in the summer he may be buoyed by lively performances from youngsters Joe Bennett, Bates and Adam Johnson.
The final chance of the game fell to West Ham's Lucas Neill who tried an audacious lob in the dying minutes but then came the dreaded whistle for Boro.
It was their 15th away defeat of the season and it will be a long summer for many of the fans shedding tears in the stands.
The Hammers, meanwhile, can reflect on an impressive season having secured a ninth-place finish in what has been an impressive first campaign for Zola.
Middlesbrough manager Gareth Southgate: "It's a low for our football club, our supporters, our players and staff.
"The club will grieve tonight but the future will start tomorrow. It's a painful blow but we have to take the pain and move forward."
West Ham boss Gianfranco Zola: "It's been a fantastic season. Today was a prefect way to finish it.
"It's been a team performance and credit has to go to everyone. We now have to build on what we have done. The structure is there so we will keep moving."
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