Saints picked up just their fourth Premier League point of the campaign away from home at Craven Cottage.
Having impressively dispatched QPR in November, Nigel Adkins' side returned to west London to eke out a 1-1 draw.
After Dimitar Berbatov opened the scoring in the eighth minute, Saints plugged away and Lambert's spot-kick five minutes from time earned a point.
"We need to be solid (away from home)," the Saints striker said.
"We have been solid in the last few weeks. We are disappointed with their goal, but I think after that they didn't create any chances.
"We are defending well at the minute. We've got to keep that up and obviously carry on scoring goals.
"We're disappointed with our performance, but I think we deserved the point.
"I think we were getting better as the game went on and we looked like we were going to score whether that penalty came or not.
"It's a good point. Away to Fulham is a good point any day of the year, so we're happy with that and we're going into Stoke away now which is going to be another hard game."
Lambert's goal was his seventh in the top flight and continues his 100 per cent penalty success rate at Southampton, although he thought Cottagers goalkeeper Mark Schwarzer had done enough to save it.
"I panicked, I honestly thought he'd saved it," Lambert said, with the relief etched on his face.
"He got a good hand on to it and it just managed to go just under the bar.
"It took me a while to realise it had hit the net, but thankfully it went in.
"I fancied myself. I always fancy myself with the penalties but I didn't quite hit it where I wanted it to go.
"I wanted it to go more in the corner, but thankfully I've hit it hard enough to squeeze in."
The strike resulted in a share of the spoils that few at Fulham could argue with.
Manager Martin Jol admitted his side did not do enough for the win and those comments were echoed by left-back John Arne Riise, who knows vast improvements are needed.
"We are disappointed not to win the game but, at the end of the day, we have to look at our performance," he told Fulham TV.
"At the moment we're not playing out best football. We're working hard to get back to the football we were playing at the start of the season.
"Did we deserve more? I don't think so. They played with a style of long balls to Lambert and made it hard for us.
"But, at the end of the day, we have to look at ourselves. We are not playing the best football so we need to get the confidence back and start playing the well we can."
The result leaves Fulham just six points off the drop zone and heightens the importance of Saturday's visit of Swansea to Craven Cottage.
"We have had some bad results lately but the team has worked hard on the training ground," Riise added.
"You could see everybody was really fired up. They really wanted to show the fans that we're back to normal.
"But they made it hard for us and even though we worked hard for the fans and ourselves to get back on track it takes some time.
"We have just got to recover now and get ready for the Swansea game, which will be another tough battle and hopefully we can get three points."