The Bosnia & Herzegovina international was caught by a challenge from Baggies midfielder Gareth Barry, and was unable to continue at The Hawthorns.
Kolasinac was subbed off in the 37th minute, with Ainsley Maitland-Niles taking his place on the flank.
The 24-year-old has made 17 appearances in the Premier League for Arsenal this season, scoring one goal and adding four assists.
Kolasinac joined the Gunners in the summer on a free transfer from Bundesliga side Schalke.
