The Ferrari driver started the race in 10th position at Silverstone but recovered to finish behind winner Nico Rosberg and runner-up Mark Webber, while Red Bull's Sebastian Vettel was forced to retire with around 10 laps remaining.
"It was a good race for us," Alonso told BBC Sport. "We got lucky to recover some points and it was a good result for us in the end. We know we have a lot of work to do and now we look ahead to the next race."
The result has seen Alonso cut Vettel's lead at the top of the Drivers' Championship to 21 points.