Can unarmed strike work with a rogue’s sneak attack?


So a situation came up last night where a player with a monk/rogue was flanking (I use the optional flanking rules to grant advantage) an enemy. He was armed with a shortsword, but even with advantage, he missed the attack roll and thus did not get to use his sneak damage.

Since he had taken the Attack action with a monk weapon (shortsword), he could make an unarmed strike as a bonus action, as granted by the monk’s Martial Arts feature. He still has advantage on that role since nobody had moved and the attack hit. But then the issue came up; is unarmed strike a finesse weapon? Is it eligible for sneak attack?

I ruled yes and allowed the sneak damage on this attack, but should I have allowed this or strictly should this not have worked?