The Cottagers needed penalties to reach round three, prevailing 5-4 following a 2-2 draw after extra time.
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 with five minutes of normal time remaining.
Substitute Michael Symes put Burton in front in the first period of extra time but Fulham sub Hugo Rodallega levelled things up again in the 117th minute and it went to penalties.
"Firstly, I think it was a great game of football," said Jol. "It was very satisfying that we kept going and we tried to grind out a result.
"That is resilience and mentality, and that is what we want. We want a winning team again.
"It could have been a bad night for us. The crowd smelt blood but we showed character and responded and it turned out nice in the end.
"We put a strong side out because we wanted to win the tie. In England the lower league sides make it difficult for you, and they made it difficult for us.
"We had all the play for about 78-80 minutes but then towards the end and in extra time they had their spell.
"You get angrier by the minute because I think we had 70% possession and you know that if you're not scoring, or if you only score one goal, then you will always have a problem.
"And it proved that way. They got their goal and then you have to go to extra time and they still had that same flow and they scored again. So it was not easy, it was difficult.
"But I am pleased with the resilience of my squad, they showed that they wanted to get through.
"In the end, if you look at the possession and control we had of the game, then I think we deserved the result.
"But I was impressed with Burton, they are a good side. They have a great home record, I think it was the best in their division last season.
"You could see that we were really worried in extra-time."
Dyer, Burton's third penalty taker in the shoot-out, saw his spot-kick saved by David Stockdale.
That miss came immediately after Bryan Ruiz had a lucky escape - his penalty striking the post but ricocheting into the back of the net off of unfortunate Albion goalkeeper Dean Lyness.
John Arne Riise then belted home the final penalty to send Fulham through.
On Ruiz's spot kick, Jol joked: "That was very fortunate. I don't like left-footers taking penalties - but John Arne Riise finished it off thankfully."
Burton boss Gary Rowett was proud of the showing of his team.
He 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."