Lazio took full advantage of defending champions Inter Milan's loss to city rivals AC Milan on Saturday to move three points ahead, but needed a late winner from Andrea Pinamonti to earn their ninth win in 10 matches for their best ever league start.
The Spanish league held a moment of silence before every match at the weekend to remember the victims of last week's deadly train crash.
Mbappe scored his 12th league goal this season to put Madrid ahead with an 11th-minute penalty, having been fouled himself. He was also involved in Jude Bellingham's goal - the England midfielder's 50th goal with Madrid - midway through the first half.
Summer signing Yan Diomande's powerful strike made it 3-0 before halftime.
Madrid are in third place, one point behind Atletico Madrid and three behind free-scoring Barcelona.
Athletic Bilbao overcame Real Sociedad 2-1 in the Basque Country derby, with Inaki Williams and Alex Berenguer scoring in the first half.
Struggling Getafe rallied to beat Sevilla 3-1, and Valencia won 1-0 at Elche.
In Italy, Pinamonti came on as a second-half substitute and ripped off his sky-blue Lazio jersey in celebration after netting the winner in the 83rd minute.
Gustav Isaksen equalised for Lazio late in the first half after Cagliari striker M'Bala Nzola had put the Sardinian side in the lead in the 21st.
The win gave Lazio their best start in club history, with nine wins and one draw in their opening 10 league games.
Elsewhere, France midfielder Lucas Da Cunha's penalty late in the first half helped fourth-placed Como to a 1-0 home win against rock-bottom Venezia.
Seventh-placed Juventus scored second-half goals to beat eighth-placed Napoli 2-0 at home.
Frosinone consolidated sixth spot with a 2-0 win over Torino.
In France, Lyon moved into third place and stayed ahead of Paris FC on goal difference after both sides won.
Goals from Ghana forward Ernest Nuamah and veteran midfielder Corentin Tolisso gave Lyon a 2-0 home win against Angers, while Mali striker Lassine Sinayoko's brace helped Paris FC beat Monaco 3-1 at home.
Strasbourg routed Auxerre 5-1 at home, while Marseille beat Toulouse 2-0.
In Germany, Niko Ilicevic made his debut in Frankfurt's 3-1 win over Hamburg at 16 years and one day old, matching Youssoufa Moukoko (for Borussia Dortmund in 2020) as the youngest player in Bundesliga history.
