There was a comfortable save for Mamardashvili as Anton Stach tried his luck from distance prior to half-time.
Four changes for Liverpool during the break included the introduction of Ryan Gravenberch for his first appearance of pre-season.
Leeds threatened as Calvert-Lewin attacked the box in the left channel, with his hit across goal evading both the far corner and the supporting Bogle.
They did halve their deficit in the 60th minute, however.
Kerkez lost possession in his own territory and Ethan Ampadu fed a pass left to Aaronson, who coolly controlled the ball and rolled it under Mamardashvili.
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In response, Federico Chiesa was foiled in an attempt to turn home a Kieran Morrison knockdown after Kerkez’s lofted cross.
Leeds soon drew level as a long throw from the left side was helped on by Joe Rodon and reached Calvert-Lewin at the far post to touch over the line.
Their swift turnaround continued with 73 on the clock as the Whites benefited from an oddly fortuitous goal.
Longstaff was primed to meet a low cutback into the area and in the process of making contact as a slight Curtis Jones touch in trying to clear sent the ball off him and bobbling into the bottom corner.
And Calvert-Lewin further extended the Leeds advantage five minutes from the end, guiding a firm header in off the crossbar from another flick-on.
**Team**
Liverpool: Mamardashvili, Kerkez (Tsimikas, 68), Wirtz (Jones, 46), Szoboszlai (Elliott, 68), Isak (Chiesa, 46), Frimpong (Scanlon, 77), Nyoni (Gravenberch, 46), Chambers (Ndiaye, 68), Morrison (Ramsay, 68), Ngumoha (Koumas, 46), Ndukwe.
Unused subs: Woodman, Davies, Abe, McConnell, Wright, Sonni-Lambie.
**Attendance:** 49,128