Gianluca Scamacca scored his first West Ham goal as the Hammers began their final European campaign with a 3-1 win over Danish club Viborg.
Italy striker Scamacca, who joined last month from Sassuolo in a £35.5million deal, scored a first-half header on his full debut.
Jarrod Bowen added a second after the break and although Viborg pulled one back, Michail Antonio completed things with the third.
It gave the Hammers some breathing room ahead of the second leg of that Europa League qualifying play-off in Denmark next week as they bid to qualify for the group stages of Europe’s third-tier competition.
West Ham were without Declan Rice and Aaron Cresswell, as well as Moyes, all banned for various indiscretions during the agonizing Europa League semi-final loss to Eintracht Frankfurt in May.
It had also been a difficult preparation for Viborg for what was probably the biggest night in their history after two players, Nigerian winger Ibrahim Said and Gambia striker Alassana Jatta, were denied entry into the country. due to post-Brexit visa rules.
Nonetheless, burly striker Jay-Roy Grot proved a handful ahead of the visitors, particularly Hammers substitute skipper Angelo Ogbonna who was making his first appearance in nine months after a cruciate injury.
But Moyes watched from the stands as nearly £50million of his summer acquisitions combined to give West Ham a 24th-minute lead.
Ex-Burnley winger Cornet made his way to the goal line, backed inside and lifted a cross for Scamacca to head home at the far post.
Viborg provided a sporadic threat and when Cornet gave the ball away Grot was sent on goal, with Alphonse Areola rushing in to save with his legs.
The second goal came in the 64th minute when Bowen collected the ball from 30 yards out, surged forward and kicked a low drive into the corner.
But four minutes later Viborg halved the lead with a header at the far post from Jakob Bonde.
Areola was forced into a mid-air save by Nils Mortimer to deny the Danes an equalizer before Antonio scored the third with a tap-in from the back of compatriot Said Benrahma.
Hearts lose in Zurich in the Europa League
Hearts dropped two soft goals in two minutes after taking the lead as they fell to a 2-1 loss to the Swiss champions FC Zurich to leave their hopes of reaching the Europa League group stage hanging in the balance.
Lawrence Shankland put Hearts ahead in the 22nd minute of the first leg of the play-off but the visitors trailed on 12 minutes.
Adrian Guerrero equalized in the 32nd minute after Hearts failed to manage a single long ball, then two minutes later Blerim Dzemaili took advantage of a more mediocre defence.
There were other 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 group action in the Europa Conference League.
Hearts will need to improve, however, to reach the top competition after Zurich brushed aside a poor start to their title defense and loss of home advantage.
Sitting bottom of the Swiss Super League with two points and a goal from five games, 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 team.
Hearts captain Craig Gordon, however, believes they have given themselves a real shot at reaching the group stage.
The guard said BBC Scotland“I thought it was a decent performance. We gave up a few goals in a silly spell, but other than that we felt pretty controlled in what we were trying to do and executed the game plan well.
“We scored our goal and were disappointed to lose two but we are well into the game coming back to Tynecastle.
“We gave ourselves a chance. We knew we couldn’t win it tonight, but we knew we could have given ourselves a mountain to climb and lost it. We’re still in the game.”