Former England global Lampard won his first game as manager 2-1 against _ thanks to a last-minute victor, but couldn't repeat the trick in his first home game in charge.
Derby had a bright start to the second half with Mason Mount dragging a shot wide but Klich set up Roofe to beat Carson with a rising drive on the hour before Alioski headed in Pablo Hernandez's cross four minutes later.
"We have got another tough game midweek, then we go to Millwall and everyone knows what it is like down there".
Following a 3-1 win over Stoke last weekend, Leeds were in good form again as former Argentina and Marseille coach Bielsa won his battle with ex-Chelsea star Lampard, who was taking charge of the Rams at home for the first time.
He said: "We were under no illusions as to how hard it would be beforehand but we didn't deal with what they threw at us".
"That was the most pleasing thing for me, not only making my debut and making over 20 appearances, but for a club like Huddersfield to stay in the Premier League".
Mat Klich put Leeds ahead with a long-range strike in the fifth minute before Tom Lawrence equalised seven minutes later.
In the aftermath, many social media users had their say about the Rams' below-par display. They moved the ball better, we made mistakes which gave them goals.
"I have spoken before about it being important to have options across all areas and Scott provides another one for us, but I think it's also important to mention that Max Lowe has been fantastic in the games so far", Lampard added. "In the first half, we were a little bit unstable, we had moments of instability so we can point out this aspect as one to correct", he told reporters.
"But the performance of the team was better today than last Sunday, the performance of some players lasted longer".