Damage dealt by Axton's Sabre Turret does not trigger this skill.
Any amount of damage dealt by Axton will trigger this skill for its full duration, including single-digit amounts dealt from jumping on top of enemies.
This skill works exceptionally well with elemental damage over time, as each "tick" resets the 3 second duration.
Igniting, corroding, or electrocuting an enemy will guarantee the player a good 10 seconds of health regeneration while the enemy is taking damage (plus the 3 seconds after damage over time wears off). The player may take cover during this period, and benefit from the health regeneration.
Throwing homing grenades over or around cover also allow a player to trigger this skill without needing to expose themselves to enemy fire, when pinned down in cover by ranged enemies. This of course works even better when combined with the aforementioned strategy, by using homing elemental grenades.