The Blues travelled to the Stadium of Light knowing a victory would have moved them within two points of Manchester City, but Enzo Le Fee's first-half penalty proved decisive.
Chelsea struggled to create clear chances throughout the contest and failed to build enough pressure to trouble a well-organised Sunderland side.
Alonso admitted his team had not done enough to deserve a better result, pointing to their slow build-up play and lack of quality in the final third.
"It wasn't the best one. I think that we didn't deserve more than probably one point," the Chelsea manager told reporters after the match.
"The game was really close. Not much happened, to be honest. Just that action. We had the cross from Morgan, but we didn't produce enough to deserve to win."
The Spaniard was particularly unhappy with Chelsea's inability to break down Sunderland's defensive structure, insisting his players needed greater patience and creativity.
"We were in the low block. We missed a bit of patience, a bit of quality, creativity in the last third to create more chances," he said.
"You need to be more consistent. We didn't do good enough."
Alonso also rejected suggestions that Chelsea had been physically bullied by Sunderland, although he acknowledged that his side could have handled key moments much better.
"No, no, not bullied. I would say that it was a situation that we could have defended better," he explained.
"It was a situation that we could have got better advantages in attack. But the game was really competitive. They were waiting for a chance. They got it after the lead."
Chelsea attempted to respond after falling behind but lacked the attacking threat to seriously trouble their opponents.
"We were trying, but we were not producing much. The first goal was so important. Being behind, we didn't produce enough," Alonso added.
The defeat leaves Chelsea fifth in the Premier League standings heading into the November international break, six points behind second-placed Brighton.
Brighton midfielder Pascal Gross has played down talk of a Premier League title challenge despite his side moving into second place with a convincing 2-0 victory over Brentford on Sunday.
Andoni Iraola was delighted with Liverpool's response to a difficult week after they returned to winning ways with a 3-1 Premier League victory over Crystal Palace at Selhurst Park.
Michael Carrick has insisted he expects to remain Manchester United manager after the international break despite the club dropping into the Premier League relegation zone.
Mikel Arteta says his side "can still be much better" following a hard-fought 1-0 win over Hull City in the Premier League on Saturday.
