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Barcelona dominated possession in the opening stages but struggled to create many clear chances, with a weak effort from Alexis Sanchez that Fraser Forster easily saved being the best of the early opportunities for the visitors.
Celtic did well to soak up the pressure and were rewarded for their stubbornness in the 21st minute when Wanyama climbed highest to head Charlie Mulgrew's corner past Victor Valdes and put the hosts ahead.
Lionel Messi almost responded within minutes of Wanyama's goal but his powerful strike was tipped onto the crossbar by Forster after the Argentinian had found space inside a crowded penalty area.
Barcelona hit the woodwork once again before the break when Alexis Sanchez headed against the post, allowing Celtic to hold on to their lead until half-time.