The full-back scored early and the striker struck late to give the defending champions an important win as they close in on World Cup qualification
Jordi Alba scored the opening goal of the game after 19 minutes before Alvaro Negredo's late effort gave Vicente del Bosque's men three precious points as they extended their lead of Group I over France to three points with two games to play.
Victory also means that Spain are unbeaten in World Cup qualifying action since slipping to a 1-0 defeat against Denmark in March 1993.
Real Madrid shot-stopper Iker Casillas quickly repaid the faith shown in him by Del Bosque to start with him ahead of Victor Valdes with a smart save with his legs to deny Finland a shock lead with just over 11 minutes gone.
Alexander Ring cut the ball back after good work down the Finland left and a Sergio Ramos defensive lunge sent it goalwards, but his club team-mate was down quickly to deal with it from close range.
Teemu Tainio's lofted ball sent Teemu Pukki clear of the Spanish defence and he crashed a volley over before the visitors went down the other end and broke the deadlock thanks to a composed Alba finish.
Fabregas picked up possession on the edge of the centre circle in the Finland half and his chipped pass found Barcelona team-mate Alba. The defender cut in from the left wing, held off a challenge from a defender and tucked it low past the onrushing Niki Maenpaa.
Villa came close to doubling Spain's lead three minutes before the half-hour mark, easily slipping his marker on the left wing before firing a rasping shot across the face of goal that flashed inches wide.
Spain dominated the remainder of the first half and showed their attacked intent within a minute of the restart as Pedro shot over after controlling Andres Iniesta's pass.
Del Bosque introduced Jesus Navas at the expense of David Villa after 55 minutes as he looked to inject some pace into the side.
Finland then created and spurned two chances in as many minutes. Tainio first rippled the side-netting with a downward header after surging to meet Roman Eremenko's corner at the far post, but neither the man on the line or Casillas looked troubled.