Edin Dzeko, last season's top scorer, rescued a point for VfL Wolfsburg in a 1-1 draw at St Pauli in Sunday's other game with his ninth goal of the season.
Podolski fired his spot-kick into the top right hand corner after Milivoje Novakovic had been fouled by Georg Niedermeier as the pair tussled for the ball.
The win, only Cologne's third in 13 Bundesliga games, took them on to 11 points, level with Stuttgart - who have a better goal difference - and one ahead of Borussia Moenchengladbach, who replaced them at the bottom.
"We were a very determined and disciplined team today," said Cologne coach Frank Schaefer.
Ulreich also blocked a drive by Petit while Cacau squandered Stuttgart's second-half chance, heading wide from two metres.
St Pauli, who had lost their previous four games, went ahead with a Markus Thorandt header in the 28th minute against former England manager Steve McClaren's Wolfsburg.
