|Cloud: I couldn't finish 'em. Looks like this's gonna get complicated.|
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Avenger is an enemy in Final Fantasy Dimensions. It is a frequent encounter in Deist Cavern.
Casts Quake upon being defeated. Quake deals around 1000 damage to your entire party. Counters harmful spells with Bio for around 600 damage and inflicts Sap. Also absorbs elemental spells and counters with Ga-tier spells; Fira will be met with Firaga, for example. Counters physical attacks with !Attack for around 400 damage to the front row. Frequently casts Libra.
A good strategy is to advance forward a little bit of the story as the Moogle in this cave suggests, returning here after the Ranger chapter is finished. Have 2 Warriors or Monks equipped with +20%HP ability, and 2 Mages that can cast Cura and Protect.
If you want to attempt the chest encounters in Deist before advancing the story, however, then a good strategy is to wait until after you have recruited Barbara; the Dragoon's Healing Breathe can help alleviate damage. Avenger highlights its target with Libra, followed by a physical attack on that target. Avenger only uses magic as a counter against magic -- you can thus limit Avenger's damage to physical attacks, which Protect and Defend can alleviate quite nicely. The F-Ability Magic Bomb 2 (Fire+Cura) does nice damage from a mage, but the resultant Bio counter will likely kill your caster at low levels (+20%HP at lower levels could allow for survival, but beware Sap). A solid physical strategy is a Warrior equipped with Focus. Focus three times and use Strike. As mentioned above characters equipped with +20%HP -- especially on a Warrior or Monk -- can survive Avenger's suicidal Quake, though your mages will likely die. Also do not attempt elemental spells. Fira, Thundara and Blizzara will result in Avenger not only absorbing these spells, but it will also retaliate with Firaga, Thundaga and Blizzaga respectively.
At higher levels (or after progressing the story) you can alternatively start the battle with Protect-All and use Focus 3 times, then use Boost Jump. Have the Mages use Defend. He will go down relatively fast. Note that Focus also applies to F-Abilities; Magic Bomb 2 is viable in this situation, even with the Bio counter.
!Flee works fine. But, it won't work against the chest encounters.