The Toffees were playing their first match since the death of Kenwright on Monday at the age of 78.
It was a performance the lifelong Everton fan would surely have been proud of with Calvert-Lewin hitting a superb second-half winner.
“Firstly, and you can’t guarantee it, but it’s befitting to win after the week we’ve had following the sad loss of the chairman,” said Toffees boss Dyche.
“The players know the depth of the chairman’s love for the club, they’re well aware of that.”
Calvert-Lewin revealed the shock of the news affected everyone at Goodison Park.
“No-one was prepared for what happened this week and it has saddened everyone at the club. That one was for Bill,” the striker told Sky Sports.
“He would have been proud of us and proud of how we won – to get the lead, dig in and work so hard. It was a very good day.”
Calvert-Lewin, who has been beset by injuries over the past couple of seasons, looks to be finally getting back to his best.
Six minutes into the second half he played a one-two with Jack Harrison, Cruyff-turned away from Kurt Zouma and Nayef Aguerd and hit a low shot past Alphonse Areola.
It was his 50th Premier League goal for Everton, joining Romelu Lukaku, Duncan Ferguson and Tim Cahill in reaching the half-century.
“Dom’s a very good player who’s had a tough couple of years,” added Dyche.
“He’s very close now, you can tell in his body language and it was a fine finish.
“He looks stronger, sharper. It is a really good marker for him to get to that number.”
A second away win of the season gave Everton a little more breathing space between them and the bottom three.
“This is a tough place to come, they’ve proved that over the last couple of years, so it was a very good performance,” said Dyche.
“We are improving from last season. I think there are clear signs of that.”
For West Ham, it was a third defeat in eight days following reverses at Aston Villa and in the Europa League at Olympiacos.
David Moyes picked an attacking line-up, handing Ghana winger Mohammed Kudus his first start in the Premier League, but their only shot on target came in the 90th minute through substitute Said Benrahma.
“That would be a problem for us,” said Moyes. “But, let’s be fair, that’s the first game we’ve not scored a goal in this season.
“But I agree, I thought we missed chances today, we missed two or three opportunities to score. I don’t think it was ever going to be a game where we had nine or 10 opportunities. I didn’t see it that way for either team.”