Erik ten Hag's side faced a frustrating wait for the breakthrough with Rasmus Hojlund and Alejandro Garnacho missing big chances at each end of the first half before Victor Lindelof grabbed the winner on 59 minutes.
Eze's sweetly struck free-kick put the visitors ahead in the 59th minute and there were grumbles from the home fans as chances were missed early in the second.
In contrast, the Sheffield United fans had little to get excited about, as their side offered little in attack other than a flurry of chances just before the break.
Joel Matip thought he had headed Liverpool into the lead in the second half, but VAR intervened and the goal was disallowed for an offside against Virgil van Dijk in the build-up.
Despite French forward Antoine Griezmann being on Simeone's list of rested players, Atletico dominated proceedings and made the most of its chances against a usually solid Las Palmas, which arrived in the capital with the second-best defensive
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Aymeric Laporte's second league goal of the season put City ahead in the final moments of the first half, Everton giving the defender the freedom of the penalty area to head past Jordan Pickford.
United appeared to be clinging on for a point as Villa created a succession of chances after Douglas Luiz had cancelled out Rasmus Hojlund's first-half strike.
It was a hard-fought and important win for Chelsea after second-pace Arsenal slipped up against Everton on Tuesday and puts Antonio Conte's side in a strong position to end the year at the top of the table.
Chelsea had the better opportunities throughout, Timo Werner wasting two good chances before Havertz struck and substitute Christian Pulisic squandering another in the second half.
Messi, making his 550th appearance for Barcelona, then added the second and third goals to ensure the Spanish champions halved the deficit at the top of the table.
Tottenham responded well to falling behind, creating a raft of chances, but Jason Steele made several good saves to deny the in-form Richarlison and Dejan Kulusevski, while James Maddison curled narrowly wide.
Chelsea will regret missing a host of chances, nearly all falling to striker Nicolas Jackson, as they suffered another agonising Wembley experience after the extra-time defeat on penalties against Liverpool in the Carabao Cup final.