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Gylfi Sigurdsson was at the centre of most of Iceland's attacks, but he curled wide from distance before placing a free kick just wide of the post.
Emil Hallfredsson was the next Iceland player to try his luck, but his 28th-minute shot also flew wide of the upright as Latvia held on to stay level after 45 minutes.
Bodvarsson was also a constant nuisance, and went close again shortly after the break.
Latvia's Artem Rudnev was shown a red card for his second bookable offence of the evening ten minutes into the second half, and the visitors went ahead shortly after through Sigurdsson, who showed great feet to engineer space for himself inside the box before curling a shot into the top corner.
Aron Gunnarsson then glanced a header into the far corner to make it 2-0 in the 77th minute, before Rurik Gislason got Iceland's third in injury time with a precise finish as Iceland stay top of the group.