Openda stepped up to the spot with eight minutes remaining but Augsburg goalkeeper Finn Dahmen, who gave away the penalty, redeemed himself by diving to his left and saving the attempt.
A "brutally frustrated" Leipzig defender David Raum said after the match "if we want to play among the big guns, we have to get through games like this."
Augsburg's Phillip Tietz gave the home side the lead late in the first half with a clean volley but Openda headed Leipzig level shortly after, his 15th goal of the season.
Slovenian striker Benjamin Sesko produced a carbon copy of Openda's header to put Leipzig ahead but Ermedin Demirovic pulled Augsburg back on level terms, blasting in with 30 minutes remaining.
Openda then had the chance to win it for Leipzig, but his spot-kick was kept out by Dahmen, who has stopped four of the 12 Bundesliga penalties he has faced.
Leipzig, who were six points clear of Dortmund at the winter break, now sit three behind having won just one in five since Christmas.
In a Saturday marked by protests against the German FA's decision to pursue foreign investment in the league, several games experienced long delays as fans threw tennis balls and foil-wrapped coins onto the pitch.
Borussia Moenchengladbach drew 0-0 at home with lowly Darmstadt in another heavily-delayed match.
Eintracht Frankfurt missed a clear chance to consolidate a European spot, drawing 1-1 at home to Bochum and will finish the weekend in sixth.
Promoted Heidenheim continued their excellent recent form, winning 2-1 at Werder Bremen.
Playing their first season in the top flight, Heidenheim moved into eighth place and have now not lost since early December.
In Saturday's late game, second-placed Bayern Munich travel to league leaders Bayer Leverkusen, hoping to inflict a first loss of the season on Xabi Alonso's side, who are unbeaten in 30 games in all competitions.
Leverkusen, who have never won the Bundesliga despite finishing second on five occasions, can go five points clear by beating Bayern.