Eddie Howe's men were punished immediately as Smith scored an own goal a minute later when Diego Costa's wayward shot deflected off him.
The Spaniard lined up a free-kick, and expertly dispatched it into the back of Artur Boruc's net.
Antonio Conte insisted Diego Costa is still an important player for Chelsea after the striker's goalless run extended to four matches as the Premier League leaders won 3-1 at Bournemouth on Saturday.
As half-time approached, Bournemouth came back into the game with Benik Afobe hitting the post in the 29th minute. "We were a bit exhausted towards the end and they're a top team, that's why they're top of the table but we'll bounce back next week".
Both sides wins means that the seven point gap between the London clubs has been maintained with just seven matches of the season left.
"The team deserve full credit because the energy and the way they played was fantastic". "Pressure brings the best out of you".
Jurgen Klopp's side's hopes of finishing in the Premier League's top four looked set for a damaging blow when Stoke took the lead through Jonathan Walters just before halftime.
The Reds sit five points below Spurs, having played a game more, after Firmino's stunning strike secured victory at Stoke.
The midfielder shot into the top corner from the edge of the area from Dusan Tadic's lay-off in the 25th minute at The Hawthorns.
"He fought a lot for the team and he is important - it is important for the team that we won against a strong team".
Sergio Aguero made it 2-0 early in the second half, polishing off Raheem Sterling's cross after a sweeping team move, and Fabian Delph lashed in the hosts' third.
Despite the defeat, Hull remained two points clear of the relegation zone, as their chief relegation-threatened rival Swansea also lost at West Ham.
Alonso bent a ideal free-kick into Boruc's top-left corner to score his fifth league goal of the season and Boruc denied Moses and Costa efforts to keep the scoreline respectable, as Conte's men moved a step closer to Premier League glory.
"I'm not feeling anything other than we need to win some more games".
Second-from-bottom Middlesbrough's hopes of beating the drop are slipping away though after they could only draw 0-0 at home to Burnley - stretching their winless run to 14 league games.