Mbappe's seventh goal since joining Madrid was sublime, with the France forward starting a give-and-go with Jude Bellingham with a heel flick, then picking up the pass inside the area and using a nifty move to clear a defender before finding the net in the 40th minute at the Santiago Bernabeu Stadium.
Lucas Vazquez and Rodrygo also scored for Madrid, who extended their unbeaten run in the league to 39 matches — with 29 wins and 10 draws going back to last season.
Alaves made a late charge in Tuesday's match by scoring with Carlos Benavídez in the 85th and Kike García in the 86th but couldn't find the equaliser despite chances in the final minutes.
"We played a great match for 80 minutes, but we suffered more than we should have in the end," Vazquez said.
The victory moved Madrid within one point of league leader Barcelona, who will go for their seventh victory in a row to start the season when they host winless Getafe on Wednesday.
Mbappe now has seven goals in nine matches with Madrid in all competitions. He has five goals in the Spanish league, one behind leading scorer Robert Lewandowski of Barcelona.
The France star, who signed as a free agent after leaving Paris Saint-Germain, also had a goal disallowed for offside in the 22nd. He was substituted by Arda Güler in the 80th.
Vazquez had put Madrid ahead one minute into the match with a shot from inside the area after a nice run by Vinícius Júnior through the left side of the area. Rodrygo added to the lead after a long run in the 48th.
Madrid coach Carlo Ancelotti, in his 300th match with the club, used most of his regular starters and did not immediately rotate his squad despite a congested schedule.
Sevilla midfielder Saul Niguez was carried off on a stretcher with an apparent leg injury early in the second half of his side's 2-1 home win over Valladolid.
And Valencia followed up its first win of the season — 2-0 at Girona — with a 0-0 draw against Osasuna.
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