Harry Kane's goal after 15 minutes proved to be decisive in London.
Alvarez, who started the game up front alongside Erling Haaland, came close to finding the net with a long-range effort, but it went just wide of the post.
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Riyad Mahrez also struck the woodwork, as Pep Guardiola's men failed to score for the fifth successive away game against Spurs. Alvarez said:
"There's a slight bitter feeling left. It was a big chance for us to close the gap at the top and pick up some points.
"We put in a good performance and played really well. It's a big chance missed so it's frustrating.
"I think we did our best, now it's all about carrying on, you try to be perfect but sometimes this happens against you.
"It is a feeling of frustration, but we're very much aware that there are lots of games from now to the end of the season.
"It's a case of picking up as many points and victories and being almost perfect from now to the end of the season."
The Argentine also gave an insight into his role alongside Haaland, as well as touching on our next league encounter against Aston Villa.
Watch the entire interview via the video at the top of the page.