Chris Erskine put the Jags ahead early at the Tony Macaroni Arena but Keaghan Jacobs lashed the Ladbrokes League One champions level as the Group H clash finished 1-1 after 90 minutes.
Former Scotland goalkeeper Alexander - making his first competitive start in over a year after completing his free transfer move from Aberdeen - then saved three of the four penalties he faced as Livi won the spot-kick decider 3-1.
There was little sign of ring rust as the teams got the new campaign under way, with both sides going full pelt from the kick-off.
Alexander settled himself into the action with two early saves from Erskine and Kris Doolan.
But the hosts were also looking to threaten and should have grabbed the lead when Scott Pittman was played in by Dale Carrick, but the midfielder’s low shot was blocked by Tomas Cerny.
However, it was the Premiership side who took command after quarter of an hour as Erskine turned Declan Gallagher one way then the other before sweeping a lovely finish into the top corner.
Both sides had goals chalked off as play continued to swing from end to end.
Thistle, though, were fortunate not to be pegged back on the stroke of half-time when Adam Barton’s slip allowed Pittman to race through, but the Livi midfielder wasted two goes at finding the net.
But there was no escape 60 seconds after the restart.
Jags midfielder Ryan Edwards thought he had done enough when he blocked Jacobs’ first strike but the midfielder collected the rebound before rifling it back with interest into the far corner.
Jacobs almost grabbed a spectacular second in the 79th minute when his long-range lob attempt forced Cerny to scramble back before tipping over.
With no more goals the game went straight to penalties but it was Livi who claimed the extra point as Alexander denied Christie Elliot, Blair Spittal and Doolan.