Enzo Le Fee's first-half penalty handed the Black Cats victory at the Stadium of Light.
The Blues pushed for an equaliser after the break, with substitutes Geovany Quenda and Danny Welbeck coming close.
However, he refused to make excuses, admitting his players lacked the quality needed to break down Sunderland's defence despite their late pressure.
“I would say there were situations that we could have defended better, situations that we could have got better advantages in attack, but the game was really even, really competitive,” said Alonso.
“They were waiting for the chance – and they got it. After they took the lead, we were trying, but we were not producing much.
“The first goal is always important and today, when we were behind, we didn't produce enough.
“It wasn't the best game. I think that we didn't deserve more than probably one point. The game was really close.”