But Zoltan Gera ended West Brom's near six hour goal drought just before the break and then on-loan Chelsea player Lukaku headed the winner with eight minutes remaining to leave Baggies boss Steve Clarke hoping the Blues do not decide to recall their player during the January transfer window.
Victory ended Albion's mini slump from which they had collected only two points from four games after climbing to third spot. Albion were also indebted to another rock solid display from central defender Gareth McAuley.
The Canaries came close to a share of the spoils despite the absence of key striker Grant Holt with a hamstring injury. Lukaku came close to opening the scoring for the Baggies after three minutes as header from Goran Popov's cross flew just over the bar.
Snodgrass connected with a volley from the edge of the Albion box but it was directed straight at Foster who returned after a seven game absence through injury. The first clearcut opening fell to Lukaku after 21 minutes when he took a pass from James Morrison in his stride.
But Lukaku failed to make a clean contact on his cross shot and Mark Bunn saved with relative comfort. It proved a costly miss as Snodgrass put the Canaries ahead three minutes later after Peter Odemwingie brought down the midfielder and a curling free-kick found the roof of the net.
West Brom tried to retaliate and a goal-bound effort from McAuley was blocked by the body of Bunn following a Chris Brunt corner, but after 43 minutes Gera brought Albion back on level terms with his fourth goal of the campaign.
Bunn parried Jonas Olsson's downward header from a Brunt corner onto the crossbar but Gera was on hand to volley home the rebound. McAuley then rescued Albion with a superb tackle inside the box before Norwich centre-back Michael Turner headed wide from Snodgrass' centre.
Clarke brought on West Brom top scorer Shane Long - in place of Gera - Then with eight minutes left Lukaku showed his power and strength when climbing above Javier Garrido to head home Popov's cross as Albion held on to claim their seventh home win of the season.