And although Gary O'Neil had briefly levelled for the hard-working visitors, the simple fact was that they had arrived in the East End for the 38th and final game of the season with just too much to do.
Not only did Southgate's 19th-placed side have to win at Upton Park, but they also needed to gain a five-goal swing over Hull City , who also had to lose against Manchester United, while neighbours Newcastle United had to suffer defeat at Aston Villa, too.
It was always going to be a tall order and, sadly for Southgate, only Sir Alex Ferguson and Martin O'Neill did him any favours, as his side failed on both counts when it came to their own double mission.
Certainly, the Boro manager's final day task had not been made any easier by the loss of Stewart Downing, who sustained a broken foot in last week's draw with Villa and the England winger was replaced by Tony McMahon.
Indeed, with just two minutes on the clock, chants of 'that's why you're going down' rang out from the short-sleeved Hammers' fans amongst the crowd of 34,007 as the unmarked Adam Johnson blasted high and wide at the far post, to set the tone for an agonising afternoon for the North-East of England.
Up in the safety zone, ninth-placed West Ham had seen their Europa League aspirations evaporate with defeat at Everton last Saturday, but with £760,000 per place prize money at stake, the East Enders still had enough incentive to try and prevent Manchester City from leapfrogging them in the table.
Gianfranco Zola made three changes as fit-again top-scorer Cole, Stanislas and Jonathan Spector came in for Diego Tristan, David Di Michele and the suspended James Tomkins.
With the visitors looking to preserve the Premier League skins, West Ham took more than a quarter-of-an-hour to muster their first effort, when Radoslav Kovac unleashed a low, long-ranger that flew into Brad Jones' rib-cage and then, midway through the half, Stanislas saw his 18-yarder loop off Robert Huth onto Boro 'keeper's left-hand angle.
At the other end, Matthew Bates, Marvin Emnes and Huth all missed the target with hurried efforts, but while news was filtering through that both Newcastle and Hull were trailing, Boro were about to be shown the art of finishing by Cole.
On 33 minutes, Luis Boa Morte found the overlapping Herita Ilunga whose low cross into the danger-zone clinically bisected both Huth and Justin Hoyte, leaving the unmarked Hammers striker the simple task of slotting home his 12th goal of the season.
That made Southgate's half-time talk even more daunting. but he responded by making a double substitution as Julio Arca and McMahon stood down for Joe Bennett and Josh Walker, who wasted no time in forcing Robert Green into his first save of the day.
The Hammers keeper stood no chance four minutes later, however, when Tuncay weaved his way into the box before playing in the supporting O'Neil, who drilled an unstoppable, low angled, ten-yarder into the far corner.
Having just taken one small step forwards, beleaguered Boro were sadly now about to take a giant leap backwards.
On 57 minutes, Stanislas collected from Kovac and let fly with a low 20-yarder, that crept between the outstretched right glove of Jones and the base of the post to condemn the visitors to the Championship.