The England midfielder scored two second-half goals — with almost identical low, side-footed finishes — in a 3-2 win on the south coast on Saturday.
Rice was a doubt for the match because of a knee injury that forced him out of the 4-1 win over Aston Villa on Tuesday, but he was fit enough to return and proved the difference between the teams.
Kicking off the year with a victory 👊 pic.twitter.com/UrWEbULbyb— Arsenal (@Arsenal) January 3, 2026
"We are going to need everyone chipping in at different points so happy to help the team," said Rice, who scored twice in a Premier League game for the first time.
Arsenal fell behind in the 10th minute when Evanilson stroked into an unguarded net after intercepting a woeful pass out from the back by Gabriel Magalhaes.
But the Brazil centre back made amends by lashing in the equaliser six minutes later following a mazy run by Noni Madueke, scoring a second goal in as many starts since returning from six weeks out injured.
After Rice's double, a long-range strike by Bournemouth substitute Eli Junior Kroupi in the 76th minute made for a tense finale but Arsenal held on to record a fifth straight win, to keep Aston Villa and Manchester City at arm's length in the title race.
Villa are the closest challengers to Arsenal after a 3-1 win over Nottingham Forest earlier on Saturday, while City can return to second place, four points behind the Gunners, if they can beat Chelsea on Sunday.
Villa rebounded from their first loss in two months — at Arsenal — when captain John McGinn scored twice in the second half, building on England striker Ollie Watkins' long-range opener in first-half stoppage time in his 250th game for the club.
Morgan Gibbs-White reduced the deficit to 2-1 in the 61st minute only for McGinn to restore Villa's two-goal cushion after a poor piece of goalkeeping from Forest's John Victor, who raced out of his area in an attempt to collect a ball over the top.Â
McGinn easily rounded Victor and slotted the ball home from 30 metres, before going off injured immediately after the goal.
Villa's 11-game winning run — which had included eight victories in the league — was ended on Tuesday by a 4-1 thumping at Arsenal.
At Molineux, Wolves beat West Ham 3-0 to end their historically bad start to a top-flight season.
No team has taken this long to record their first win in a Premier League since England's top-flight competition was rebranded in 1992.
Jhon Arias, Hwang Hee-chan and Mateus Mane scored first-half goals for Wolves, who moved on to six points, still 12 from safety with 18 games remaining.
Wolves manager Rob Edwards said he was torn between feeling "a bit of relief and really pleased."
"We should enjoy it," he said, "but it's only one win. That's all it is."
Wolves are only six points behind next-to-last Burnley, who lost at Brighton 2-0, and eight off third-to-last West Ham.
