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Podolski netted twice before the half-hour mark and although the Gunners took their foot off the gas in the second period, Giroud and Cazorla came off the bench to make the scoreline emphatic in the final stages.
The Gunners were completely dominant in the opening minutes against their League One opponents, although they broke the deadlock with a goal which came after a slick counter-attacking move.
Another of his German compatriots, Per Mertesacker, provided Podolski with his second, flicking on a corner at the near post for the versatile forward to nod home from close range on 27 minutes.
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Arsenal's goalscorer and Serge Gnabry had chances to put the tie to bed before the break but were wasteful, and Coventry came out after the interval and nearly worked their way back into the match.
Their only real effort on goal in the first period came through Carl Baker's solo attempt, a run and shot from distance which forced Lukas Fabianski into a diving save over his crossbar, but they were more dangerous after the resumption of play as Leon Clarke had a couple of one-on-one opportunities.
He firstly took too long to get a shot away and fired straight and Fabianski, and then hit the post on 57 minutes following good work from Franck Moussa, as complacency crept into Arsenal's performance.
However, their substitutes changed that as Giroud netted on 84 minutes, drilling into the bottom corner from Kieran Gibbs's layoff, and Cazorla rounded things off five minutes later by volleying in a rebound.