Antonio Conte's men quickly doubled their advantage as N'Golo Kante released Eden Hazard with a lovely long ball and the Belgian worldwide rounded Artur Boruc before slotting home for 2-0 in the 20th minute.
But Bournemouth, who had already hit the post, got back into the contest through Joshua King's long-range deflected effort.
But Alonso settled the visitors' nerves with 22 minutes to go, the Spanish wing-back arcing a sumptuous free-kick over the wall and into the top-right corner, leaving Boruc rooted to the spot.
Tottenham kicked off the day's action at White Hart Lane and for 33 minutes it looked as though Watford could frustrate Mauricio Pochettino's side's ambitions of putting pressure on Chelsea.
- Chelsea have conceded in nine consecutive Premier League games in a single season for the first time since the 2011-12 season (nine games between Aug-Oct 2011).
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. That pressure, it's normal.
Liverpool sit five points below Spurs, having played a game more, after Firmino's stunning strike secured victory at Stoke.
No Teesta pact during Hasina visit this time
The Bangladesh Prime Minister will also honour 1661 Indian soldiers who lost their lives during the 1971 Liberation War, he said. Modi will host a luncheon for Hasina at the Hyderabad House where Mamata is expected to be present.
Jonathan Walters headed a close-range goal from Xherdan Shaqiri's cross to give Stoke a 44th-minute lead, prompting Reds boss Jurgen Klopp to make changes at half-time.
Bournemouth made a determined start to the second half too, with Fraser shooting narrowly wide.
The players already knew the Spurs result heading into the game, which added to the pressure and "test of character" posed by a relentlessly positive and fearless AFC Bournemouth. "You need to win games like this". "Now it is important to rest and restart and prepare for the game against Manchester United".
City took the lead just past the half hour when Jesus Navas' cross was turned into his own net by Ahmed Elmohamady, then Sergio Aguero doubled the lead three minutes into the second half.
Alonso bent a flawless 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.
"It was important for us to win after many days without a win", said Guardiola, whose side lost 2-1 at Chelsea in mid-week after three successive draws.