The German has struggled for form and confidence since arriving at the Emirates Stadium last summer after departing Chelsea in a £60 million deal.
The jury is still out on whether or not his move to Arsenal will bear fruit, but getting off the mark will undoubtedly bolster his confidence.
Arsenal travelled to the south coast as one of only four sides yet to taste defeat in the Premier League this season, even though they had struggled to get going at the top end of the pitch.
By contrast, Bournemouth welcomed the unbeaten Gunners boasting the Premier League’s longest current winless streak (D3, L7).
It was always going to be a tricky tie for the Cherries and they were pegged back early as Bukayo Saka, who was a doubt for the match, headed into an empty net in the 17th minute.
The Gunners maintained their dominance, doubling their lead on the stroke of half-time as Martin Odegaard slotted home from the penalty spot after a rash challenge from Max Aarons on Edward Nketiah.
Havertz was handed his headline moment eight minutes after the restart when Ryan Christie conceded a penalty, and the German cooly slotted home what he will hope is the first of many.
A late headed goal from Ben White helped the visitors ease to a victory, marking just the third time Arsenal have started a Premier League campaign with three consecutive away wins.
It was also the first time since 2021 that the Gunners had achieved the feat, which takes them back into the top four in second place, one point behind Manchester City, who suffered a shock defeat to Wolverhampton Wanderers.
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