The champions dominated Saturday's Premier League clash at Goodison Park but failed to make the most of their opportunities against Sam Allardyce's diligent side.
The result, coupled with Manchester City's 4-0 win over Bournemouth, means Antonio Conte's side are 16 points adrift of top spot with half the season gone.
Alonso acknowledged it was a "very disappointing" result and has urged his team-mates to be more ruthless against Brighton and Hove Albion on Boxing Day.
"We were very unlucky not to win after controlling the whole game and having a lot of chances," he said, as quoted by the club's official website. "But we take one point, it is what it is, very disappointing.
"We knew it was going to be tough and as the game was passing it was getting harder because they were defending closer to their goal. The performance wasn't bad, we just need to be more clinical."
Never deserved to draw today but we keep going! Roll on Tuesday! #EVECHE @ChelseaFC— Marcos Alonso (@marcosalonso03) December 23, 2017
Merecimos los 3 puntos hoy pero el balón no quiso entrar! A trabajar para el próximo partido! #CFC pic.twitter.com/TWHiewxs3b
Centre-back Antonio Rudiger felt luck was against Chelsea, who had 68 per cent of the possession and 25 shots on Saturday, but wants to respond by ending 2017 with wins over Brighton and Stoke City.
"Of course, it's very disappointing because I think we were the better team and we had a lot of chances to score but you have some days when it's like that," said Rudiger. "You can even play for maybe three hours and you wouldn't score.
"We needed the three points so we have to keep going. We played a good match but we were just unlucky.
"The target can only be six points. It's not easy but that's our aim, that's what we're going for, we want six points."