Carlo Ancelotti's side, who had thrashed Mallorca 6-1 on Wednesday night, are now just three points ahead of Sevilla, who have a match in hand.
Alaves had lost their opening five games, but went in front after just four minutes through Laguardia's header, who cleared a shot off the line during the second half to help secure all three points.
Youssef En-Nesyri's early strike and a late goal from Rafa Mir helped Sevilla beat Espanyol 2-0 at the Estadio Ramon Sanchez Pizjuan.
Sevilla midfielder Thomas Delaney was sent off for two cautions in as many minutes during the second half.
Substitute Marcos Andre struck deep into stoppage time to grab 10-man Valencia a 1-1 draw at home against Athletic Bilbao.
Inigo Martinez broke the deadlock with 20 minutes left, and after Maxi Gomez saw red late on, it looked like no way back for Los Che.
However, with five of the seven additional minutes up, Andre drilled the ball home from close range for his first goal since his summer transfer from Valladolid.
Paris St. Germain maintained their 100 per cent Ligue 1 record with a 2-0 victory over Montpellier at Parc des Princes.
Despite missing Lionel Messi through injury, PSG were still too strong for the visitors, as Idrissa Gueye fired them in front early on and substitute Julian Draxler added a late second.
PSG move 10 points clear of Marseille, who have played two matches less.
Jonathan David scored twice, his second from the penalty spot, to give champions Lille a 2-1 win at Strasbourg, who had Adrien Thomasson sent off late on.
Nice coasted to a 3-0 win at strugglers St. Etienne.
Amine Gouiri set them on their way with an early goal before Calvin Stengs and Andy Delort wrapped things up in the second half as the hosts suffered a fifth straight defeat to sit bottom.
Saturday's other evening kick off saw Lyon held to a 1-1 home draw by Lorient, as an early second-half goal from Karl Toko Ekambi cancelled out Armand Lauriente's opener.
With Bayern Munich having won on Friday night their rivals lost ground, Wolfsburg beaten 3-1 at Hoffenheim.
Bote Baku had given the visitors the lead, but Andrej Kramaric equalised on the stroke of half-time before Christoph Baumgartner completed the comeback in the final 20 minutes.
Leipzig returned to winning ways with a 6-0 thrashing of Hertha Berlin at the Red Bull Arena, Christopher Nkunku scoring in each half.
Arminia Bielefeld's search for a Bundesliga victory goes on after they lost 1-0 at Union Berlin to a late goal from substitute Kevin Behrens, while it finished 1-1 between Eintracht Frankfurt and Cologne.
AC Milan went top of Serie A with a 2-1 win at Spezia.
Daniel Maldini, 19-year-old son of Paolo and grandson of Cesare, headed the visitors into the lead early in the second half.
Daniele Verde equalised with 10 minutes to go.
However, but Brahim Diaz's close-range finish secured all three points for Rossoneri, who edge ahead of Napoli ahead of their game against strugglers Cagliari on Sunday.
Defending champions Inter sit third after they were held to a 2-2 draw by Atalanta.
Edin Dzeko scored an equaliser in the 71st minute – which ended the Nerazzurri's 18-game winning run at home.
Saturday's late match saw Nikola Kalinic scored in the 90th minute to earn Verona a dramatic 3-3 at Genoa who had fought back from 2-0 down to lead with a late double from Mattia Destro.