The Southampton striker was introduced midway through the second half at Wembley and took just two minutes to make an impact, brilliantly heading home Leighton Baines' corner with his first touch to seal a 3-2 win.
It capped a madcap few days for Lambert, whose call into Roy Hodgson's squad last week came on the same day his wife gave birth to their third child.
"That's what I've been dreaming of probably all my life," said the 31-year-old.
"It means so much.
"It was a corner, obviously I fancied myself. What a ball by Leighton - exactly where I wanted it to go - and I managed to get rid of my marker and get a firm header.
"I've tried to enjoy it, but my focus was to try and do well. I'm thankful for it to have gone like that.
"The atmosphere was superb. For that to be my first (England game) is unbelievable."
Lambert said he thoroughly enjoyed his time with the England camp and thanked his team-mates for helping him to settle into his new surroundings.
"I was trying to play it cool as if it wasn't bothering me but deep down I wanted to scream," he said. "It was brilliant.
"I've got to say a big thank you to the lads who made me feel welcome. I had respect for them anyway but being with them for a few days I've got a lot more respect."