The forward struck either side of a David Cano goal after Mauricio Pochettino named a strong squad for the game in Palafrugell, Catalunya, with new signings Victor Wanyama and Dejan Lovren included alongside regular first-teamers such as Adam Lallana and Luke Shaw on the bench.
The ex-Burnley man had the ball in the net as early as the 12th minute but was denied by the offside flag.
However, Rodriguez continued to wreak havoc and was rewarded 16 minutes later when he capitalised on a mix-up between Llagostera keeper Jose Moragon and defender Pere Segarra to slot home the opener.
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The 23-year-old almost doubled the visitors' advantage on the hour-mark when he ran on to a through ball from strike partner Lambert, only to be denied at point-blank range by Llagostera's substitute goalkeeper, Nico Ratti.
And the home side took full advantage of this reprieve to draw level just two minutes later when defender Cano nodded Adriano Garcia's corner beyond Paulo Gazzaniga.
However, Rodriguez was in no mood to be upstaged and he restored the Premier League outfit's lead on 67 minutes. Ratti saved well from Lallana's shot but the striker was on hand to convert the rebound.
That proved to be the man of the match's last contribution, as he was taken off for young midfielder James Ward-Prowse.
Wanyama then went close to his first goal in Southampton colours as the former Celtic midfielder let fly with a vicious half-volley that went just wide two minutes from time as the Premier League team's superior fitness and quality shone through in the closing stages. However, the Saints could not add to their lead and had to settle for a 2-1 victory.