Grady Diangana gave West Brom a 10th-minute lead with their first goal of the season, firing past Jordan Pickford from distance.
Everton equalised through Calvert-Lewin's clever backheel finish on 31 minutes, an offside decision against him being overturned by the VAR.
James Rodriguez's superb maiden strike on his home debut put Everton ahead before Kieran Gibbs was sent off for serious foul play on the Colombian on the stroke of half-time.
Two minutes after the break, Matheus Pereira's brilliant free-kick pulled West Brom level, only for Michael Keane to restore Everton's lead on 54 minutes.
Calvert-Lewin tapped in his second on 62 minutes and completed his treble four minutes later, making it the first time in eight seasons that Everton have started a campaign with back-to-back wins.
West Brom are still seeking a first point of the season.
King of the Match Dominic Calvert-Lewin
Everton: 26 Sep v Crystal Palace (A)
West Brom: 26 Sep v Chelsea (H)