The Brewers took the Premier League side all the way to penalties at the Pirelli Stadium on Tuesday night.
The Cottagers finally prevailed 5-4 on spot kicks after a 2-2 draw following extra time.
Martin Jol's men looked comfortable with the first-half lead given to them by Adel Taarabt but Jack Dyer, who saw his penalty saved by David Stockdale in the shoot-out, equalised after 85 minutes to send the tie into extra time.
Substitute Michael Symes put Burton in front but Fulham sub Hugo Rodallega levelled things up again with three minutes remaining and it went to penalties.
Rowett said: "I thought we had some brilliant performances.
"To almost knock them out was incredible. I thought we were brilliant, absolutely brilliant.
"We've got a brilliant group of players here and over 90 minutes we are still technically unbeaten this season."