Brave Cowdenbeath made Premiership Thistle sweat well into extra time before the full time Premiership side made it into the last 16 of the League Cup.
Keeper Grant Adam was their star with several outstanding saves while Jamie Stevenson will reflect on what might have been after missing a penalty when the game was scoreless.
Thistle manager Alan Archibald made six changes to the side that lost for the first time in the Premiership last Saturday but visiting gaffer Colin Cameron turned out virtually the same side that had lost all three Championship games this season.
Beath had the better of the opening stages with home keeper Scott Fox alert in the sixth minute to deflect for a corner a shot by Andy Russell, who had been set up by Greg Stewart.
However, once the Jags picked up the pace, and allied it to their long insightful passing, the visitors spent most of the first half in defence.
Less than two minutes in to the second half Adam pulled off a marvellous save, getting down to a shot from Baird cutting in from the left.
Cowdenbeath then had the chance, in 55 minutes, to take the lead when clever footwork by Kenny Adamson in the home box provoked a clear penalty with the offender Aaron Muirhead taking his pins away.
The Fifers could not take advantage. Jamie Stevenson tried a fancy run-up but home keeper Fox was up to it, diving to his left. The well marshalled Beath defence thereafter saw the game through to extra time.
Two minutes into extra time Thistle sub Christie Elliott seemed to have won the game when he side footed home, but a minute later Beath replacement Kane Hemmings brought equality again.
However further Thistle counters by Steven Lawless and Aaron Muirhead, a penalty, saw the Firhill side through.