Gibson capitalised on a loose clearance from a corner to give the Premier League club the lead with eight minutes gone before Anichebe rose highest to head a Kevin Mirallas cross past Ian Dunbavin from eight yards out with 13 minutes on the clock.
Mirallas added a third moments before the interval with a dipping free-kick from some 30 yards out but Peter Murphy finished off a brisk move down the right to reduce the arrears.
A raft of second-half changes saw Antolin Alcaraz make a debut but impacted on Everton’s fluency - and they only added one more goal in a fractured second-half with Leon Osman arriving late to head a Seamus Coleman cross past Dunbavin just after the hour mark.
Jay Rodriguez got both goals as Southampton beat Llagostera 2-1 in Girona.
Rodriguez capitalised on a mistake from Llagostera's keeper Moragon to give his side the lead with 28 minutes gone after the keeper miscontrolled a backpass before Mauricio Pochettino made 10 changes at half-time – including the introduction of Victor Wanyama.
David Cano headed past Paulo Gazzaniga from a corner with 61 minutes gone to level things up but Rodriguez re-established the lead six minutes later when he followed up a saved Adam Lallana effort.
Charlton were held to a 1-1 draw against AFC Wimbledon at the Cherry Red Records Stadium.
In-form winger Danny Green gave the visitors the lead with a low drive midway through the first-half but an Alan Bennett header with 31 minutes gone was enough for the League Two club to ensure a draw.
Andrea Orlandi grabbed a 13th minute winner as Brighton & Hove Albion beat Getafe at the Coliseum Alfonso Perez.