Villa were beaten on aggregate by League Two Bradford in the Capital One Cup semi-final on Tuesday and then saw his side exit the FA Cup at the hands of Millwall on Friday night.
The latter defeat came in the cruellest way possible as, with the tie seemingly headed for a replay, Villa conceded an 89th minute goal at the Den.
However, Lambert vowed his side would bounce back - and insisted he would be at the helm to oversee their revival.
"I'm not a quitter and I'll keep going," Lambert said. "You take the knocks - but you have to come back stronger.
"I thought we dominated the game - the very least we should have got was a replay.
"We bossed it really, with the possession. It's individual errors that are costing us on set-pieces and the second goal was just a crossed ball and we should deal with that.
"I thought we showed grit. I thought we dominated the game and I never thought either team looked as if they were going to win it towards the end. It looked as if it was going to a replay."