according to rules 

 "In other words, the attacking player must
assign damage equal to the number of hit points
each defending unit possesses to that unit before any
damage can be assigned to the attacked section on
the defending player’s capital. Note that more damage
can be assigned to a unit at the attacker’s discretion,
in anticipation of the Toughness keyword or other
damage cancellation effects, but a minimum damage
equal to the number of remaining hit points possessed
by each defending unit must be assigned to
each defender before any damage can be assigned to
the defender’s capital....

.....The defender may also
over-assign damage to any attacking unit, in anticipation
of Toughness or cancellation effects."

if i got it right, if i want to kill a unit with toughness 1 i just have to assign 1 more damage to it than the hit points the unit has.

so the toughness saves a unit only if the opponent has an amount of damage equal to the hit points of the unit to assign to it?

Yes. Just think of Toughness X as an armor that soaks the first X damage thrown at its way. I suspect only a few unit would be able to survive damage from repeated combat, so for all intends and purposes, Toughness can be treated as a direct increase in hit points in a one-off scenarios.

thx. that makes clear