It took Everton just two minutes to take the lead at The Hawthorns, with Romelu Lukaku producing an emphatic finish against his former loan club following Jonas Olsson's scuffed clearance.
The visitors' second goal arrived in the 66th minute and again came via a West Brom error.
Kevin Mirallas raced away from the hosts' defence and fired towards the bottom corner, seeing his effort squirm underneath Ben Foster, who should have made a comfortable save.
After four Premier League matches, winless West Brom sit second-from-bottom on two points and Irvine acknowledged that his side were making things difficult for themselves unnecessarily.
He told reporters: "You're talking about two international players [Foster and Olsson], two proven Premier League players and they don't normally make those kinds of mistakes.
"For us to make one, let alone two, in one game is unusual. But these are lads who are generally very consistent in their performances and good players.
"They are devastated about the goals and mistakes happen. We just have to accept that and move on.
"They certainly didn't mean to make those mistakes, that's for sure. They're devastated at the moment and they'll be hurting for the next few days I'd expect.
"You talk all the time about making sure you get off to a great start and we've got off to the worst possible start again today. Having said that, we recovered from that today and in the first half we played well.
"We got forward well, got into some good positions and got a lot of crosses into the box. We didn't particularly play with the quality that is required at the top of the pitch and lacked cutting edge."
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