Skipper Sean Wilson was at his brilliant best, leading by example with a paid maximum from six outings to inspire his side to a lead which should make them favourites for the return at Purfleet on September 19.
Hammers boss Ronnie Russell took full advantage of the new Cup regulations by nominating three of his riders for their points to count double.
But the meeting started on a sour note when promising youngster Daniel King was filled in with dirt and could not see where he was going towards the end of Heat Two. He lost control and hit the safety fence at high speed and was taken to hospital with a suspected fractured wrist, leaving brother Jason to take over the rest of his rides for the Hammers.
Sheffield used rider replacement for ankle injury victim Andre Compton, a facility which brought just six points to the Owlerton outfit - half of what Compton would normally be expected to score on his home circuit.
Arena, meanwhile, also operated the facility for injured Dane Henning Bager and Hammers managed only four points, all from Kelvin Tatum as a golden double in Heat Eight.
Russell seemed to run out of options after using Joonas Kylmakorpi in Heat Nine, the Finland World Cup star looking strangely out of sorts on a track where he performed so well in the league match back in July.
It wasn't until Heat 14 when Russell introduced a tactical substitution, Leigh Lanham going off a 15-metre handicap as a golden double, but he couldn't catch impressive Tigers youngsters Andrew Moore and James Birkinshaw who extended the home side's advantage.
Wilson completed a brilliant night with a hard-fought win over Lanham and Tatum in Heat 15, diving between both riders in a spectacular move to pull clear down the back straight at full throttle.
Sheffield will be delighted with the outcome - but they know they could have had an even bigger lead had Moore stayed on his machine when handily placed in his first ride and hadn't suffered a retirement through mechanical problems in this third outing.
Tigers: Sean Wilson 17+1, Ross Brady 12+1, Ricky Ashworth 9, James Birkinshaw 8+1, Andrew Moore 7, Richard Hall 4+1.