The result has put paid to the club's hopes of Champions League football next season.
And it also severely dented their chances of a Europa League spot after Southampton leapfrogged them into sixth place with a win over Tottenham Hotspur.
Despite leading the East Londoners to what is officially their best ever Premier League season, Bilic cut a frustrated figure as his side were just not at the races in what was only their seventh defeat of the campaign.
Some West Ham fans showed their displeasure at full-time as the frustration at seeing their side fail when it mattered most once again - they lost the FA Cup quarter-final replay to Manchester United last month.
'It'll be alright on the night'
But now Sinclair, who played for the Hammers more than any other club during his career, has reassured supporters ahead of a crunch European football six-pointer against the Red Devils on Tuesday night.
"Will all end happily at Upton Park on Tuesday night #UnderTheLights #whufc #mcfc," Sinclair posted on Twitter.
Europa League hopes dented
In a career spanning five years at Upton Park, Tricky Trev - as he affectionately became known on the terraces - scored another 35 goals in 177 appearances.
If the Hammers do win in the last game at their beloved home of 112 years before the big move to the 60,000 seater Olympic Stadium, they then have to match Southampton's result on the final day to stand any realistic chance of qualifying for the Europa League.