José Mourinho's side came into the fifth matchday looking to respond after their previous league defeat to Real Betis, and they made a blistering start in front of their home supporters.
Real Madrid were awarded a penalty in the 14th minute after Álvaro Carreras was brought down inside the area. Mbappé stepped up and calmly converted to put the hosts ahead.
Carreras then doubled the advantage just three minutes later. The full-back exchanged passes with Mbappé inside the Rayo penalty area before finishing into the corner to make it 2-0.
Madrid continued to dominate and added a third in the 35th minute. Bernardo Silva and Vinícius Júnior combined on the left before Vinícius delivered a cross towards the far side of the area, where Jude Bellingham finished with a first-time strike.
Rayo had barely threatened during the opening 45 minutes, but they emerged with greater attacking intent after the interval.
The visitors pulled one goal back in the 51st minute when Sergio Camello capitalised on a mistake in the Real Madrid build-up to reduce the deficit to 3-1. The goal briefly gave Rayo hope of an unlikely comeback.
Real Madrid, however, gradually regained control. Thibaut Courtois made important saves as Rayo searched for another goal, while goalkeeper Augusto Batalla also kept the visitors in the contest with several impressive interventions.
The hosts eventually put the result beyond doubt in added time, with Mbappé completing his brace after receiving a pass from Arda Güler before producing a composed first-time finish.
The goal was Mbappé's seventh in six appearances this season and capped another influential performance from the France international.
Match result
• Real Madrid: Kylian Mbappé 14' (pen), Álvaro Carreras 17', Jude Bellingham 35', Kylian Mbappé 90+1'
• Rayo Vallecano: Sergio Camello 51'
The victory was Real Madrid's fourth consecutive win at the Santiago Bernabéu and moved them onto 12 points from five matches, level on points with Barcelona ahead of Barcelona's game against Levante.
Although Madrid were significantly less convincing after half-time, their explosive first-half display ultimately proved decisive. Mbappé's continued scoring form, Carreras' attacking contribution and Bellingham's influence gave Mourinho's side enough quality to secure a comfortable victory.
Real Madrid will now turn their attention to their next LaLiga fixture away to Elche on September 15, before the Madrid derby against Atlético Madrid later in the week.