Gianluca Scamacca scored his first West Ham goal as the Hammers began their latest European campaign with a 3-1 win over Danish club Viborg.
Italy striker Scamacca, who joined last month from Sassuolo in a £35.5m deal, netted a first-half header on his full debut.
Jarrod Bowen added a second after the break and although Viborg pulled one back, Michail Antonio rounded things off with the third.
It gave the Hammers some breathing space ahead of the second leg of this Europa Conference League qualifying play-off in Denmark next week as they bid to make the group stages of Europe’s third-tier competition.
West Ham were without Declan Rice and Aaron Cresswell, as well as Moyes, with all banned for various indiscretions in the agonizing Europa League semi-final defeat to Eintracht Frankfurt in May.
It had also been a tough build-up for Viborg for what was probably the biggest night in their history after two players, Nigerian winger Ibrahim Said and Gambian forward Alassana Jatta, were denied entry into the country due to visa rules post-Brexit.
Nevertheless, burly striker Jay-Roy Grot was proving a handful up front for the visitors, especially for stand-in Hammers skipper Angelo Ogbonna who was making his first appearance in nine months after a cruciate injury.
But Moyes watched on from the stands as almost £50m worth of his summer acquisitions combined to give West Ham a 24th-minute lead.
Former Burnley winger Cornet worked his way to the byline, cut back inside and lifted over a cross for Scamacca to head home at the far post.
Viborg were providing a sporadic threat and when Cornet gave the ball away Grot was sent through on goal, with Alphonse Areola racing out to save with his legs.
The second goal arrived in the 64th minute when Bowen picked the ball up 30 yards out, surged forward and pinged a low drive into the corner.
But four minutes later Viborg halved the lead through a far-post header from Jakob Bonde.
Areola was forced into a flying save from Nils Mortimer to prevent the Danes from drawing level before Antonio scored the third with a tap-in from fellow sub Said Benrahma’s pull-back.
Hearts lose at Zurich in Europa League
Hearts lost two soft goals in two minutes after taking the lead as they fell to a 2-1 defeat at Swiss champions FC Zürich to leave their hopes of reaching the Europa League group stage in the balance.
Lawrence Shankland gave Hearts the lead from the spot in the 22nd minute of the play-off first leg but the visitors were behind inside 12 minutes.
Adrian Guerrero equalized in the 32nd minute after Hearts failed to deal with a simple long ball, and then two minutes later Blerim Dzemaili capitalized from some more poor defending.
There were further scares but Hearts emerged from their first European game in six years with a real chance of progressing in Edinburgh next Thursday. The losers are guaranteed Europa Conference League group action.
Hearts will need to improve though to reach the more prestigious competition after Zurich brushed off a poor start to their title defense and the loss of home advantage.
Sitting bottom of the Swiss Super League with two points and one goal from five matches, Zurich had to make a 50-mile trip to St Gallen with their own stadium booked for a pop concert but they overcame that to beat the Scottish Premiership side.
Hearts captain Craig Gordon, however, feels they have given themselves a real chance of reaching the group stage.
The goalkeeper told BBC Scotland: “I thought it was a decent performance. We gave away a couple of goals in a daft spell but apart from that we felt fairly controlled in what we were trying to do and carried out the game plan well.
“We got our goal and were disappointed to lose two but we are well in the game going back to Tynecastle.
“We have given ourselves a chance. We knew we couldn’t win it tonight but we knew we could have given ourselves a mountain to climb and lose it. We are still in the game.”