Cobb's heroics pushed the Steelbacks up to 273 for six, but it was nowhere near enough as Pakistan chased it down to win by eight wickets.
Cobb was named County captain for the game and after winning the toss, he opted to bat.
Pakistan struck with the very first ball of the innings as Richard Levi dragged a full, wide ball from Mohammad Amir back on to off-stump.
Ben Curran and Cobb steadied things, pushing things along to near 50 before Curran too dragged one on and went for 23.
Ricardo Vasconcelos’s stay at the crease was a short one as a sharp bouncer from Mohammad Hasnain hit him on the side of the helmet and forced him to retire hurt an over later.
Vasconcelos went through a concussion assessment with first-team physio Barry Goudriaan and while the keeper-batsman was cleared, he took no further part in the match.
Thurston and Cobb’s partnership reached 48 before Thurston pushed one back to the bowler and went for 25.
Saif Zaib also departed caught and bowled for one.
Hampshire loanee Ian Holland and Tom Sole both made starts, Sole sending one sailing into the supporters club before he was trapped in front for 22.
Brett Hutton, making his return from injury, and Cobb then put on the most enterprising partnership of the innings.
In his first game of the season, Hutton found the boundary four times to finish with 27 not out from 32 balls.
Cobb was the man of the hour though, batting for all but one ball of the innings and posting his seventh List A century.
Cobb finished 146 not out from 145 deliveries, finding the rope 12 times and striking six maximums.
The Steelbacks finished with a respectable total, but Pakistan were able to make light work of it.
The visitors flew out of the gates to post 50 after just six overs.
Fakhar Zaman reached his century in a stand that passed 150, but the opener fell shortly after as a clever slower ball from Nathan Buck crashed into the stumps.
Pakistan would lose their second wicket after the score passed 200, Holland taking his first for the club as he bowled Imam-Ul-Haq for 71.
But Pakistan eventually reached the victory total with nine overs to spare.