And instead it was Derby who were able to level matters in the 12th minute after Liam Cooper fouled Lawrence on the edge of the box.
Lampard felt that Derby didn't show the quality in possession that they had shown throughout pre-season, and it cost them against a Leeds side that impressed during their 3-1 win against league favourites Stoke City last week.
Lampard will now be charged to do better than that and take Derby back to the top flight for the first time since 2008, when the club was relegated with just 11 points.
Hernandez made a great run to pull the ball back from the right for Alioski to head past Carson, and Saiz should have had a fifth for the visitors but miskicked with the goal at his mercy.
The pace of Leeds was causing Derby problems and they were nearly in again in the 28th minute when Saiz powered through the middle, but his 20-yard shot was straight at Carson.
The dip and swerve on the ball gave Bailey Peacock-Farrell no chance in the goals. Carson was called into action to deny Roofe at point-blank range with Tomori making a last-ditch block to deny Ayling on the rebound.
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The away side managed to hold on to that lead until half time.
Lawrence got in a cross that hit Craig Bryson and went behind but Leeds nearly scored a third in the 56th minute when Saiz found Roofe.
Ezgjan Allioski added a fourth in a double salvo just three minutes later.
A 4-1 lead should have then become 5-1 but Saiz sliced an effort wide after a Cooper through ball.
With the signing complete, Malone could be in line to make his debut for the Rams when they host Leeds United on Saturday evening and after a late winning goal in their opening game against Reading, Lampard will be hoping his side can carry their winning momentum into their second game.
Derby County: Carson, Wisdom, Keogh, Tomori, Lowe, Ledley (Johnson 45), Bryson, Mount (Bennett 70), Wilson (Jozefzoon 61), Lawrence, Waghorn. Subs not used: Blackman, Jansson, Roberts, Harrison.