The team website quoted the Spaniard as telling his team not to be downhearted after he and Brazilian Felipe Massa finished 14th and 15th respectively at Silverstone on Sunday.
"OK guys, keep your heads up. We will win the championship," Alonso said over the team radio after setting the fastest lap right at the end of the race.
"We are only halfway through the season and there is plenty of time to recover. We have a great car and we proved that to everyone this weekend and confirmed it with this fastest race lap.
"It is only the first half and we know we have to do better in the second one and we will do exactly that," he added.
Ferrari's performance on Sunday was the Italian team's worst of the season and their first failure to score any points in a race since the 2009 season-ender in Abu Dhabi.
Massa has now failed to score points in his last three races while Alonso had taken a mere 19 in that period and is now fifth overall, 47 points behind McLaren's championship leader Lewis Hamilton with nine races and a maximum 225 points left.
Ferrari are third in the constructors' standings, 113 points behind leaders McLaren.
"We are angry that we did not get the results which we were capable of delivering, but this anger must be turned into a positive force," team boss Stefano Domenicali said at a debriefing on Monday.
"We just have to knuckle down and continue with the development of the car, working calmly, with great concentration and determination. Then, the results will come," he added.
Trebles all round for Hamilton and McLaren
Spotted: Lewis Hamilton in Bahrain
Massa likely to stay: Ferrari
Hamilton wins Canadian GP in one-two for McLaren
Ferrari curse bad luck in Montreal