Puel, a summer replacement for Ronald Koeman, finally saw his side produce a dominant display.
Prague’s best opportunity fell to leading scorer David Lafata, who forced Fraser Forster into a neat low save after a 20-yard shot.
Substitute Jay Rodriguez put the icing on the cake in the second minute of stoppage-time, coolly side-footing home a cross from Shane Long.
SOUTHAMPTON (4-1-2-1-2): Forster; Martina, Van Dijk, Yoshida, Targett; Romeu; Ward-Prowse, Hojbjerg; Tadic; Long, Austin. Subs: McCarthy, Fonte, Bertrand, Davis, Clasie, Redmond, Rodriguez.
SPARTA PRAGUE (3-5-2): Koubek; Holek, Costa, Kadlec; Karavaev, Frydek, Sacek, Vacha, Sural; Kadlec, Julis. Subs: Bicik, Mazuch, Marecek, Cermak, Lafata, Holzer, Pulkrab.
Referee: Manuel Grafe (Germany).