England goalkeeper Ramsdale joined the Gunners from Sheffield United last week as part of a deal that could eventually be worth £30million.
Bernd Leno started in goal against Chelsea on Sunday, but Ramsdale comes in for the second-round trip to The Hawthorns.
Odegaard also starts after getting visa clearance following his £30m move from Real Madrid last week, while club captain Pierre-Emerick Aubameyang leads the line after recovering from Covid-19.
Arsenal are without Ben White (Covid-19), Eddie Nketiah (ankle) and Thomas Partey (ankle), but Alexandre Lacazette makes the bench after recovering from Covid-19.
Arsenal XI: Ramsdale, Tavares, Chambers, Holding, Kolasinac, Elneny, Xhaka, Odegaard, Pepe, Saka, Aubameyang
Subs: Leno, Mari, Maitland-Niles, Sambi, Smith Rowe, Martinelli, Lacazette
Tonight 8.00pm BST, Sky Sports
Venue: The Hawthorns