Roberto Martinez’s side once again failed to transfer impressive European results to the Premier League, in which they have taken 11 points from their last 15 matches.
Yet they could justifiably claim to have deserved something from the match against an Arsenal side who wobbled to victory.
Giroud’s selection proved pivotal to that win, as did Wenger’s decision to give centre-back Gabriel Paulista a first Premier League start in place of the out-of-form Per Mertesacker.
Gabriel was shaky at first, misjudging a long clearance and allowing Lukaku to get in behind him, with Ospina racing from his line to pat the ball clear.
But the Brazilian defender won applause from the home fans for a perfectly timed challenge as Everton’s Belgium striker threatened to tear through and score eight minutes before half-time.
That lifted an Arsenal side who had been ponderous for the opening 35 minutes, and they were soon ahead.
Giroud, who had earlier planted a header wide from eight yards out, found a few inches of space inside a crowded penalty area to turn a lovely low volley into the corner of the net from Ozil’s corner.
Arsenal had to survive a significant spell of Everton pressure after half-time, but aside from Lukaku’s effort, the only chance came when substitute Aaron Lennon steered Seamus Coleman‘s low cross straight at Ospina.
Wenger’s side did finish strongly, and only a magnificent saving tackle by Phil Jagielka stopped Ozil scoring from Alex Oxlade-Chamberlain’s through-pass.
Rosicky eased the jitters of the home fans with a shot that was deflected past keeper Tim Howard by the unlucky Jagielka.