The Three Lions duo found the net in the first half as Brendan Rodgers’ side came from a goal down to lead 2-1, before an 88th-minute equaliser from their Bulgarian hosts denied them victory.
Lambert told Sky Sports: "It was my first start in the Champions League and it was good to score.
"We have answered some of our critics about Sunday’s defeat against Crystal Palace. It was about the win and we almost got it. Now we have to go for an even better performance on Saturday.
"We are devastated about the goal at the end but it was about the performance today – win, lose or draw – and the performance was better.
Jordan Henderson was also on target for Liverpool
Ludogorets took the lead after just three minutes, but Lambert’s second goal in as many matches and his first in the Champions League leveled the scores moments later.
The England striker reacted quickest and headed home after the Ludogorets' defence failed to clear Henderson’s ball into the box.
Half an hour later Henderson turned from provider to scorer as he linked up with his international team-mate Raheem Sterling to put Liverpool 2-1 in front.
Sterling beat his man on the left, before whipping a low ball across the area which Henderson latched on to and tapped home from four yards out.
England full-back Glen Johnson also played 90 minutes during the 2-2 draw, as Liverpool earned a point which maintains hopes of reaching the knock-out stages.
Henderson added: "We knew it would be a tough game, they are a very good team technically but we showed great character to stay in the game, we looked dangerous on the counter and were more solid tonight but disappointed to concede late on.
Alex Oxlade-Chamberlain rues his luck after hitting the woodwork
"Bravery is about getting on the ball as well - having the confidence to do that when you are a goal down and I felt we did that.
"There were a lot of positives tonight but we still have a long way to go to get where we want to be."
Arsenal enjoyed a more successful night as they recorded a victory and booked their place in the last-16 with a 2-0 win over Borussia Dortmund.
Three Lions midfielder Alex Oxlade-Chamberlain made his 100th appearance for the Gunners at the age of just 21, and almost marked the occasion in perfect style when his stunning 25-yard volley crashed against the woodwork in the 55th minute.
England defenders Calum Chambers and Kieran Gibbs also turned out for Arsene Wenger’s side. The pair played the full 90 minutes and helped Arsenal keep their first clean sheet in four games.
