Cesc Fabregas' first-half goal turned out to be the victor and came after Championship team Derby had twice equalised after going behind to own goals.
Instead of a foul Zappacosta took a quick throw-in and the late arriving Fabregas tucked in a few moments later.
Lampard received a warm welcome from the home supporters as banners displayed the words "forever a blue, forever a legend" and his name was sung by Chelsea supporters shortly after Fabregas' goal.
Hazard is looking to win the Premier League title for the third time in his Chelsea FC career, having previously got his hands on the top-flight crown in 2015 and 2017.
In his programme notes before the game, the Italian acknowledged that he has so many players performing well that it is becoming hard to narrow them down to a starting XI.
Chelsea denied granting Derby permission to field Mount and Tomori was a favour to Lampard, who scored 211 goals in 648 appearances over 13 years, saying it was more a test of character for the players. So for me to try and jump forward to Chelsea or anywhere else... This could be the first game of many [for them] at Chelsea.
Zola defended the duo with regard to Chelsea's sloppy defending - but pulled no punches in his assessment of their poor distribution.
Kevin Anderson to face Kei Nishikori in Vienna final after beating Verdasco
The Spaniard broke Anderson twice in the second but didn't get another chance on the South African's serve in the final set. The eighth-ranked Anderson defeated Kei Nishikori 6-3, 7-6 (3), handing his opponent his ninth straight defeat in a final.
"The defenders had more time, but they weren't as sharp as normal", he said.
"Tomori and Keogh have been absolutely outstanding for us all season".
Tomori's own goal after just five minutes - which deflected backwards off his standing leg as he attempted to clear Davide Zappacosta's cross - was greeted with encouragement from Lampard, who gestured to remain calm.
Chelsea's loanees were involved in goals at both ends.
"We've put ourselves in great credit, coming to Stamford Bridge and to Old Trafford, but we are back to league action now and we mean business".
The last time these two sides met, Chelsea emerged victorious with a 3-1 win at Pride Park in the League Cup in 2014.
Key Stat: Ruben Loftus-Cheek has scored twice as many goals in his past two games than he did in his first 35 for Chelsea. It is home. Not my current home, but my home for such a long time. But now they have gone out and they can concentrate on the Championship, which is what is so important now.
Lampard's Derby beat Manchester United and with luck, could have come out on top against another top-six Premier League team.