A sublime second nearly arrived in the 31st minute - Rakitic adding further to an accomplished showing by gliding towards the box and exchanging passes with Messi to draw a superb save from Pau Lopez.
Messi set Barcelona on their way with the opening goal after 26 minutes, although looked to have been offside.
Messi completed his treble midway through the second half by finishing off a sweeping team move, lashing home another pass from Alba from inside the area.
Barcelona are the only team with a 100% record and are yet to concede a goal, while La Liga champions Real Madrid are sixth after a 1-1 home draw against Levante earlier on Saturday.
Barcelona scored their third in the 67th minute, however, when a clever pass from Alba found Messi unmarked inside the Espanyol box, and the 30-year-old completed his hat-trick with a brilliant finish into the far corner.
As expected, it was Barcelona that took control of the early possession, but aside from a Messi effort, which flew harmlessly over the crossbar, Espanyol restricted the hosts to very little inside the opening 15 minutes.
Barcelona: Marc-Andre ter Stegen 7 (out of 10), Nelson Semedo 6, Gerard Pique 8, Samuel Umtiti 7, Jordi Alba 8, Gerard Deulofeu 7, Ivan Rakitic 7, Sergi Busquets 7, Andres Iniesta 7, Lionel Messi 9, Luis Suerez 7; Paulinho (for Busquets, 73) 6, Andre Gomes (for Iniesta, 77) 6, Ousmane Dembele (for Deulofeu, 68) 8.
Espanyol's Pablo Piatti was unlucky not to pull a goal back when his sweetly-struck left-footed volley came back off the post just before half-time.
Suarez was back in the Barcelona line-up despite an injury scare while on worldwide duty but was not enjoying himself in front of goal as much as Messi.
Bale missed a gilt-edged chance just before, while in injury time Toni Kroos came inches from a victor with an effort from the edge of the area which struck the outside of the far post.