Arsenal got three goals in the first 20 minutes of the match, and cruised from there as the Gunners confirmed their third-place finish.
The 26-year-old Walcott got things going early with the opener just inside four minutes with a very impressive strike from a tight angle, slamming the ball past West Brom defender Jonas Olsson and off the underside of the bar into the back of the net.
Walcott, earning just his fourth start of the season and first since March 14, snatched his second just 10 minutes later as Arsenal’s fast start continued. Wilshere burst into the box, touched centrally to Mesut Ozil and a quick touch came to Walcott who skipped past a defender and poked the ball home while falling over.
It would get better for the Gunners as another returnee, Wilshere, picked up Arsenal’s third still prior to the 20-minute mark. An off-balance shot from Alexis Sanchez was blocked and fell to Gabriel Paulista, who touched to Wilshere outside the box and the England international slammed a thunderbolt into the top corner.
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Theo forced a massive save from Boaz Myhill to keep out his hat-trick in just the 23rd minute, but he would get the triple on 37 minutes. Wilshere and Santi Cazorla played a one-two on the left before the latter cut a cross along the face of goal and Walcott was there to poke it home at the far post for his third of the day.
West Brom got one back as Gareth McAuley headed home a corner when David Ospina missed with a punch, but that was it for the Baggies. They nearly had another in stoppage time when Ospina failed to track a Craig Gardner shot properly but he just was able to tip it over.
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Arsenal finish the season in third with 75 points, 12 back of the champions Chelsea. West Brom ends the season in 13th with 44 points.
Arsenal: Ospina, Bellerin, Gabriel, Mertesacker, Gibbs, Coquelin (Ramsey 69′), Cazorla, Wilshere (Oxlade-Chamberlain 77′), Ozil, Alexis, Walcott (Giroud 69′).
Goals: Walcott 4′, 14′, 37′; Wilshere 17′
West Brom: Myhill, Dawson, McAuley, Olsson (Anichebe 51′), Lescott, Fletcher, Yacob (Baird 60′), Morrison, McManaman (Gardner 54′), Brunt, Berahino.
Goals: McAuley 57′