Teenage debutant Stanislas bundled Gianfranco Zola's side into an interval lead on his first start for the club, before Tomkins' second-half header cemented a victory that leaves sorry Sunderland just three points clear of the drop-zone.
Unbeaten in their previous four league matches, eighth-placed West Ham made two changes from the side that had drawn at Blackburn Rovers a fortnight ago, as 19-year-old Stanislas and Matthew Upson came in for Radoslav Kovac (thigh) and Scott Parker (groin).
Sitting in 15th spot, Sunderland arrived at Upton Park without a win in their past five games and, following defeat at Manchester City last time out, Ricky Sbragia made four switches as Craig Gordon, Djibril Cisse, Tal Ben Haim and Kieran Richardson replaced the injured George McCartney and substitutes Marton Fulop, Kenwyne Jones and Carlos Edwards.
Having just seen Blackburn Rovers leapfrog them with a lunch-time victory over Tottenham Hotspur, the Black Cats showed their claws early on as Grant Leadbitter launched a 30-yard rocket that flew just wide and then the homegrown midfielder played a neat one-two with Cisse before seeing his shot brilliantly blocked by Robert Green.
Midway through the half, Steed Malbranque also sent a low 18-yarder just an inch-or-so wide of the base of Green's left-hand upright.
Without 11-goal Carlton Cole, who picked up a groin injury during England's win over Slovakia, last Saturday, West Ham were struggling to find any way to goal.
Although Tomkins had glanced Mark Noble's corner wide in the opening seconds, Zola's Italian-Spanish strikeforce of Diego Tristan and David Di Michele simply did not have the pace or bite against a visitors defence superbly marshalled by ex-Hammer Anton Ferdinand.
As the half-hour approached, though, Di Michele sent an 18-yard curler just a whisker wide as the sun-soaked home fans amongst the crowd of 34,761 suddenly saw their side start to rise from their slumber.
Still Sunderland looked more threatening and, just before the break, Cisse's sizzling 20-yarder was superbly palmed aside by the flying Green.
But just as a tedious opening period, notable only for the battle for midfield supremacy, looked set to end goalless, Stanislas slid the Hammers into an unlikely lead from six yards, after Luis Boa Morte rode David Bardsley's lunge before unselfishly squaring to the youngster.
Buoyed by that unexpected interval lead, Zola's men doubled their advantage seven minutes into the second period, after Tristan mustered his first shot of the afternoon.
Gordon performed wonders to divert the Spanish striker's goalbound 18-yard shot aside, but when Noble floated the resulting corner towards Tomkins, the Scottish shot-stopper was left helpless as the defender's downward header squirmed over the line.
The Hammers' delight turned to despair midway through the second half, when Jonathan Spector was stretchered off following an aerial clash with Ben Haim, while Sunderland introduced Jones and Edwards at the expense of Daryl Murphy and Malbranque in a bid to get back into the contest.
Bardsley, Cisse and Richardson each responded by forcing Green into awkward saves before the Hammers' keeper beat Jones' angled effort aside, to leave the Wearsiders looking over their shoulders near the foot of the table.