Los Blancos succumbed to a solitary second half strike by Argentine midfielder and Man of the Match Alexis Mac Allister in a match where the visitors hardly threatened goalkeeper Giorgi Mamardashvili.
Madrid only made seven shots at goal in the whole match, with only Jude Bellingham and Arda Guler's first half attempts being on target.
"It was a demanding game, really fast paced, said Alonso after the match.
"We competed well but we lacked a real threat. The first half was fairly evenly matched and the second was decided by small margins.
We started giving away too many free-kicks and the odd corner. Courtois kept us in it there. The goal gave them the edge and then we struggled a bit after that.
The defeat ended the Spanish giants perfect start to the 2025/26 UEFA Champions League campaign following three wins against Marseille, Kairat and Juventus.
"I can't ask for any more from my team," added the Spanish head coach.
"We tried our best. It's the opening phase of the Champions League, we'll have chance to recover those points, and we have to think about what's coming up."
Real Madrid currently occupies fifth spot on the league phase table and will face Greek giants Olympiacos next on 26 November following the FIFA international break.
Manchester City secured a second successive Premier League victory at home against Sunderland, Chelsea were held scoreless away to Bournemouth while Tottenham managed to beat Brentford.
Aston Villa delivered a major jolt to the Premier League title race as Emiliano Buendía struck deep into stoppage time to seal a 2-1 win over Arsenal on Saturday.
David Moyes urged Everton to stay on the front foot as they prepare for Saturday's Premier League meeting with Nottingham Forest at Hill Dickinson Stadium, where Sean Dyche made his first return since leaving the club nearly a year ago.
Hansi Flick has urged Barcelona to maintain their momentum as they travel to Sevilla to face Real Betis today aiming for a fifth straight La Liga win, and the German has used the build-up to highlight the growing influence of Eric Garcia.
