A goal from Leighton Baines gave Everton a slender lead in their Europa League third qualifying round tie against Slovakian side MFK Ruzomberok on Wayne Rooney's return.
A frustrating display looked set to end in a draw until Baines cracked in a deflected shot from 20 yards.
Rooney was quiet when playing up front but improved once he moved back behind the striker in the second half.
The second leg takes place in Ruzomberok next Thursday.
Everton were dominant in possession throughout but still needed Maarten Stekelenburg to make a crucial save and turn Jan Maslo's header on to the crossbar.
Ronald Koeman gave debuts to summer signings Cuco Martina, Michael Keane, Davy Klaassen and Sandro Ramirez but will need a better performance if they are to reach the play-off round in August and avoid a repeat of him not reaching the group stage with Southampton two years ago.
All eyes on Rooney
It is 13 years since the then-teenage Rooney left Everton for Manchester United for £27m, with his move back this summer causing almost as much discussion and debate as his departure.
Is he the same player who left? No. But will he prove to be a shrewd signing for Koeman? For free, you'd think so.
He will be judged against stiffer opposition than the side who finished third in the Slovakian Fortuna Liga last season, but he will also have to play better to even keep his place in a rejuvenated Everton side.
Rooney has spent much of the last few seasons in midfield but started in the centre of a front three, showing some neat touches but also reminders that his sharpest days are perhaps behind him.
He had the best chance of the opening half hour, but his touch was not great and he could not get any power behind a shot, and he and Everton greatly improved once Sandro came on in the second half.
With Sandro stretching the Ruzomberok defence, Rooney and Klaassen looked dangerous in pockets behind him. Finding the right position for Rooney may be Koeman's biggest challenge in the coming weeks.
"It's something I've been looking forward to for the last few weeks," Rooney said.
"It's obviously great to be at Goodison wearing the blue shirt, it was a great feeling.
"But it was an important game for us so I had to put my emotions to one side. I've done that now and can get down to focusing more on the football."
First glimpse at new-look Everton
What is not in doubt is that Koeman is in the process of assembling the strongest squad at Goodison in years. Ross Barkley may be set to leave but Everton continue to be linked to Swansea midfielder Gylfi Sigurdsson as his replacement.
Even without Sigurdsson, Koeman now has plenty of attacking options to suggest losing Romelu Lukaku's 20+ goals a season may not be a disaster, although another out-and-out striker would be a shrewd addition before the end of August.
Sandro's arrival changed Everton's game, but Klaassen was the pick of the new signings on show, going close with a shot in the first half and a header in the second, all the time knitting together the best of Everton's play on a performance which was very much in pre-season mode.
But considering the injured Yannick Bolasie, Barkley, Ramiro Funes Mori and Seamus Coleman all missed out, £30m goalkeeper Jordan Pickford was on the bench with Sandro, Tom Davies, Gareth Barry, Phil Jagielka and Ademola Lookman and Aaron Lennon, and Muhamed Besic and James McCarthy were not involved at all, it is clear this is a Toffees squad packed with quality.
Whether it is good enough to challenge for a top-four place is another question, so perhaps winning the Europa League remains the best chance for Everton to qualify for the Champions League.
And for all the attacking talent on display, it was left-back Baines who found the breakthrough, scoring with a deflected shot from the edge of the area after a corner was only headed out to him.
Man of the match - Davy Klaassen (Everton)
'We can get another number nine'
Everton boss Ronald Koeman: "It's always difficult against these kind of teams who drop back. We are not 100% yet, it is pre-season and it is a good result. We would like to score more, we had chances but every clean sheet in European football is a good result.
"When Sandro came on Wayne Rooney had more freedom and we had pace up front which made the team better. We know that we would like another striker, you normally need two nines in the squad. I am really confident that we can bring in a good one to make us better.
"It is up to the board if we go back in for Sigurdsson, everyone knows our interest but I don't get involved in the finances."
- 29Calvert-LewinSubstituted forRamírezat 61'minutes
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- Georgi Kabakov
- Shots on Target