Calvert-Lewin, who scored the winner in England’s Under-20 World Cup final win, broke the deadlock in the 80th minute with a low angled strike from 12 yards.
Everton had barely threatened prior to the goal with Morgan Schneiderlin going closest with a header.
Maybe Dominic feels a bit more comfortable as one of two strikers
Everton manager Ronald Koeman
Dominik Kruzliak headed over for Ruzomberok early in the match.
Relive the action as it happened
That and a strike from Czech winger Erik Daniel, which whistled past the Everton post, were the only efforts to trouble keeper Jordan Pickford, who was making his debut following his £30m summer move from Sunderland.
At the other end, keeper Matus Macik also had a quiet evening, with Everton – who won the first leg 1-0 – rarely troubling the Slovak defence.
Tom Davies was the only player to stand out in the Toffees attack.
He twice picked out team-mates with good passes, however both Wayne Rooney and Michael Keane wasted their chances when in good positions.
Everton’s best effort prior to the goal fell to French midfielder Schneiderlin, who should have done better with a header from Sandro Ramirez’s corner.
The breakthrough finally came with 10 minutes of normal time remaining. Another summer signing, Davy Klaassen, played in substitute Calvert-Lewin on the right who struck a low first-time effort past the reach of Macik.
The draw for the play-off round takes place on Friday at 12:00 BST. The ties will be played on 17 and 24 August, with the 22 winners going into the Europa League group stage.
Everton manager Ronald Koeman praised Calvert-Lewin, suggesting which role might suit the 20-year-old in the team.
“Maybe Dominic feels a bit more comfortable as one of two strikers. He’s strong, he’s fast and he scored a great goal,” said the Dutchman.
“It was a physical game and 90 minutes that the players needed. We are through to the next round and I’m really happy.”
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