The PHB on p. 195 (or the corresponding section of the basic rules) says:

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

The rules do not make a distinction between "main-hand" and "off-hand" weapons.

My character dual wields a shortsword with her right hand and a dagger with her left hand.

Could my character attack with her left hand dagger first, doing 1d4 + STR/DEX mod damage, then attack with her right hand shortsword, doing 1d6 damage?

Does it matter which weapon I attack with first when two-weapon fighting?

up vote 15 down vote accepted

There is no difference which hand you attack with first in general

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative. (PHB 195)

There is no distinction between a main and an off-hand in 5e unlike in previous editions. Thus, attacking with either left or right hand first does not matter.

With different weapons in each hand it could matter, but only in the case where one weapon offers a higher ability bonus to add to the attack (since only the first attack gets the ability bonus added to it). In that case, you would want to use that hand/weapon first.

In your case specifically, it doesn't matter because both the weapons are finesse and thus it doesn't matter which one you add your Dex/Str bonus to, it will be the same. Thus, she could absolutely attack with her left hand dagger first, doing 1d4 + Str/Dex mod damage, then attack with her right hand shortsword, doing 1d6 damage.

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