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The former Mancheseter United striker had already emerged as the favourite to take over from Malky Mackay, who was sacked by Tan in December.
Interim boss David Kerslake will be in charge for the Arsenal clash after overseeing the 2-2 draw with Sunderland on Saturday. However, Solskjaer watched from the stands ahead of being formally appointed.
After his illustrious Old Trafford playing career, the 40-year-old made his first steps into management at Molde.
He took over in 2011 and led them to two Norwegian Tippeligaen titles and a Norwegian Cup crown.