So as long as you qualify for Sneak Attack, it applies. In your example, the melee attack would trigger Sneak Attack.
You can even get sneak attack on an opportunity attack later in the round because it's only limited to once per turn. Sneak attack is not an attack action, it's a triggered event based on its prerequisites.
The official Sage Advice Compendium confirms this:
Can you use green-flame blade and booming blade with Extra Attack, opportunity attacks, Sneak Attack, and other weapon attack options?
Introduced in the Sword Coast Adventurer’s Guide, the green-flame
blade and booming blade spells pose a number of questions, because
they each do something unusual: require you to make a melee attack
with a weapon as part of the spell’s casting. First, each of these
spells involves a normal melee weapon attack, not a spell attack, so
you use whatever ability modifier you normally use with the weapon. (A
spell tells you if it includes a spell attack, and neither of these
spells do.) For example, if you use a longsword with green-flame
blade, you use your Strength modifier for the weapon’s attack and
Second, neither green-flame blade nor booming blade works with Extra
Attack or any other feature that requires the Attack action. Like
other spells, these cantrips require the Cast a Spell action, not the
Attack action, and they can’t be used to make an opportunity attack,
unless a special feature allows you to do so.
Third, these weapon attacks work with Sneak Attack if they fulfill the
normal requirements for that feature. For example, if you have the
Sneak Attack feature and cast greenflame blade with a finesse weapon,
you can deal Sneak Attack damage to the target of the weapon attack if
you have advantage on the attack roll and hit.