Divine Smite's damage is extra damage on the same attack, so it can kill the zombie.
The paladin's Divine Smite feature description says:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage.
It's extra damage on the same attack - it's not a separate occurrence of damage, and happens simultaneously with the weapon's damage. You can also tell that it's not "sequential" because it says it can be used when the weapon hits, not after the weapon's damage is applied.
(Some DMs may houserule that the paladin can wait to see whether the regular damage kills an enemy before using Divine Smite so that it's not wasted, but nothing in the description of the feature supports this.)
The zombie's Undead Fortitude trait says:
Undead Fortitude. If damage reduces the zombie to 0 hit points, it must make a Constitution saving throw with a DC of 5 + the damage taken, unless the damage is radiant or from a critical hit. On a success, the zombie drops to 1 hit point instead.
Since the damage from Divine Smite is part of the same attack it's applied to, the attack it's used on will kill the zombie if it reduces the zombie to 0 HP.