Everton loanee Conor McAleny squandered a decent chance to put the hosts ahead with less than a minute on the clock, having fired wide of the target when in a good position.
After nearly falling behind early on, the Cottagers did well to steady themselves and they soon found their passing rhythm as the opening quarter of the contest wore on.
Chances proved to be kept to a minimum, however, as both sides laboured away without truly troubling at either end in the London derby showdown.
Fulham upped the pressure in search of an opening goal, and they very nearly found it when Jamie O'Hara fired in a shot from range which Pope made a real hash of.
Kit Symons' charges did finally find a way through shortly after the half-hour mark thanks to Tunnicliffe, who capitalised on another error from Charlton's keeper to rifle home Ross McCormack's parried free kick.
There was a chance for Karlan Ahearne-Grant to level things up at the end of the first half, but the opportunity passed the Addicks forward by to see Fulham head into the break sitting on a one-goal lead.