Chelsea led through Fernando Torres' early goal before Bayern levelled through Franck Ribery and then Blues midfielder Ramires was sent off.
However, Chelsea regained the advantage in extra-time through Eden Hazard only for Javi Martinez to equalise for Bayern with the last kick of the game.
Romelu Lukaku was the only man to miss in the shoot-out as the roles were reversed after Chelsea had beaten Bayern on penalties in the 2012 Champions League final.
"It's hard because it was the last kick of the game, literally," Lampard told Sky Sports. "I think with 10 men our desire to take the lead and stay in the lead was unbelievable.
"It's frustrating. I thought we deserved to win it on pure desire to stay in the game.
"When you're playing Bayern with 10 men for a big slice of the game, and you go and take the lead again, and we hit the bar and had chances. I know they had chances, but they always will do. I think we played a great game and that's why it's hard to take."
Chelsea have made a bright start to the Premier League season and experienced midfielder Lampard is hoping that his side can maintain their form.
He added: "It's a long season and a lot will happen. We can take bits from tonight and improve even more, but we know when we're at our best we can challenge anyone."
Petr Cech produced a string of fine saves to keep Chelsea in the game and he was close to stopping Xherdan Shaqiri's effort in the shoot-out.
The Chelsea keeper said: "That's the way it goes. We gave them heartache in the Champions League final when we scored in the last minute, they missed a penalty in extra-time and in the end lost on penalties.
"Today, it was the reverse. We conceded with the last kick of the game and the luck fell on them in the penalty shoot-out.
"It's a cup to win. We always said we wanted to win and we respected the cup. We had a great attitude towards it.
"We did ever so well down to 10 men to get back in the game and to concede in the last second is always tough to take."