Mark Noble scored twice while Sebastien Haller and Felipe Anderson were also on target as the Hammers stopped the rot that had set in under previous manager Manuel Pellegrini.
Moyes received a lukewarm reception from the home fans before kick-off, but those underwhelmed by his appointment were not complaining by the end.
He said: “I don’t think I could have had a better start. But I can’t take the credit, the players take the credit with their attitude and commitment. It felt as if they could give the supporters something. They played really well.
“We’ve only had a couple of days with them and I tried to make it as simple as I could. In truth, I don’t know most of the players but I tried to use every bit of time. I had a conversation with them all and spoke to one or two individuals.
“Ultimately we tried to work on a good defensive shape and making sure we didn’t lose possession.
“It’s the first game and I felt I didn’t have enough time to work with the players. That’s why it’s difficult to take any credit.
“It was about being brave on the ball, trying to regain confidence, trying to regain the trust from the crowd.”
West Ham took a 17th-minute lead when Robert Snodgrass exchanged passes with Anderson and reached the byeline before pulling the ball back for Noble, pictured, on the edge of the area.
The midfielder’s first-time shot probably would not have troubled Aaron Ramsdale but for a deflection off Lewis Cook which wrongfooted the Cherries goalkeeper.
Nine minutes later West Ham doubled their lead when Ryan Fredericks overlapped down the right and crossed towards Haller. The French striker, who had looked drained of confidence under Pellegrini, showed a glimpse of why he cost £45 million in the summer with an acrobatic volley past Ramsdale.
The Hammers were in uncharted territory by the 35th minute when Noble scurried into the area and tumbled to the floor under Harry Wilson’s challenge.
Noble appeared to have tripped himself over but a VAR check deemed he had still been fouled, and the Hammers skipper tucked away the penalty.
Anderson then grabbed a breakaway fourth after the interval to finally get off the mark for the season.
Moyes added: “I’d been asking him if he’s a Brazilian. Brazilians score goals, Brazilians make the difference, they do stuff that makes us all excited.
“All I’d said to him was ‘give us a Brazilian goal’. And to be fair to him, he did.”
Everything was going West Ham’s way, so much so that Aaron Cresswell even had a red card for a foul on Ryan Fraser downgraded to a yellow by VAR.
While the Hammers levered themselves back out of the bottom three, injury-ravaged Bournemouth slid into it.
ManagerEddie Howe said: “It’s been a really difficult run.
“We’ve been knocked quite a lot this season with the injuries and the run that we’ve had. We had players playing who were not 100 per cent fit.
“It’s not an excuse. The players gave everything but they were not in the best physical condition to play. At this level you get found out.
“It’s our first time in the relegation zone for a long time, but that might be a good thing for us. It will change the mindset.”