The hosts completed a 266-run win in the third Test at the Ageas Bowl, with Moeen Ali starring on the final day as he took a further four wickets to finish with figures of 6-67.
Cook has faced intense scrutiny over his captaincy after England had gone 10 Test matches without a victory and the opening batsman was out of sorts with the bat.
Knocks of 95 and 70 not out helped lift the pressure on the 29-year-old in Southampton and the opening batsman was able to reflect on an impressive all-round display from his side following the disappointment of their second Test defeat at Lord's.
"It was the perfect game," he said at the on-pitch presentation. "It was a great performance from day one. We pretty much won every session, well we didn’t lose a session, and that’s credit to the guys. Every one of the XI has contributed to a great win."
Cook offered particularly praise to off-spinning all-rounder Moeen following his maiden five-wicket haul in Test cricket.
"His bowling has come on leaps and bounds from the beginning of the summer," Cook added. "He's worked really hard on it with the guys in the nets, telling him about the lines to bowl, which are a bit different in international cricket.
"He's responded incredibly well and on a spinning wicket to get a six-for to win the game you couldn't ask for any more."
Having struggled for form for so long, Cook was elated to have contributed with the bat.
"I said at Lord's that I thought my game was heading back in the right direction and with a 10 and a 20 it's probably a strange thing to say but to back it up I was incredibly pleased," he continued.
"I was frustrated not to get a hundred in the first innings but I'll take 160 runs in a game."