The Basque side moved up from seventh to sixth, displacing Alaves to go level on points with fifth-placed Valencia who downed Real Madrid 24 hours earlier.
Mesa put Sevilla into a 41st minute lead with a perfectly weighted rising shot from the right that dipped over keeper Fernando Pacheco and into the far corner of the net.
With Alaves struggling to make an impact the home side doubled their advantage when Sarabia’s skimming shot flashed past the hapless Pacheco and into an empty net with 10 minutes remaining.
For Sevilla this was a vital three points after Valencia had inflicted Zinedine Zidane’s first defeat since returning as Real boss on Wednesday.
Valencia’s 2-1 home success moved them to within a point of La Liga’s top four, on 46 points, only ahead now of Sevilla on goal difference.