The England midfielder struck twice as his side came from behind to beat Aston Villa 2-1, in the process taking his career tally for the club to 203.
"It means everything, it's amazing," he told Sky Sports 2.
"The support I've had from team-mates, you saw that at the end and I'm thankful to every one of them because they're the ones that put it on a plate for me sometimes.
"And the fans, they've been with me for a long time and were probably frustrated alongside me not getting there so I'm just delighted."
Asked which of his goals meant more - the equaliser which drew him level with Tambling or the winner to break the record - he said: "Breaking!
"Bobby Tambling is a great man, he's not been well lately but he's a great man.
"I was pleased to level it but I didn't want to overcook that celebration out of respect for him. But to then go and break it, I was delighted.
"I've been here a long time, I lost my mum a few years ago and she was my biggest supporter so it's for her."
He admitted Saturday ranked among his finest in a Chelsea shirt, rivalling last season's Champions League win in Munich.
"It's right up there," he said. "Not much beats Munich because that was amazing for the team and the club but personally it's another great day."
Manager Rafael Benitez told Sky Sports 2: "I said before, hopefully he can score two goals then we don't need to talk about Frank (any more).
"It's a fantastic achievement for him to score the number of goals he has, hopefully he can score more in the next two games."
Lampard's brace came after both Ramires and Villa goalscorer Christian Benteke had been sent off.
The win virtually clinches third place in the Barclays Premier League for Chelsea, who now lead Arsenal by five points and Tottenham by six with those two teams each having two games to play.
And Lampard said: "We showed character, to go down to 10 men - I know it evened up but you still need character to keep going and we got through it.
"It's huge for the team, with the games coming up next week and particularly the (Europa League) final midweek, we wanted to put it to bed today.
"It's a massive game Wednesday, we need to rest and be ready for that because it's a very tough game.
"At Chelsea we're all about trying to win titles, we've done it for a long time now."
Chelsea's win could have been more comfortable had the officials judged that either Gary Cahill or Demba Ba's efforts had crossed the line in the same second-half incident.
But Benitez said: "I haven't seen the replay. They told me from another angle it was close, but we cannot blame the referee."
Opposite number Paul Lambert was more certain, saying: "I've just seen it, the whole ball's not over the line. They got that one right, but they'd 90 minutes to get another few right. I thought some of the decisions were bizarre."
Lambert was referring to both Benteke's second booking, for catching John Terry with a high boot, and the decision not to send Terry himself off for a foul on Yacouba Sylla when referee Lee Mason ruled that Ashley Cole was covering to prevent a goalscoring opportunity.
The Scot added: "My own team were absolutely fantastic, they gave everything they had and certainly didn't deserve that.
"They were fabulous, you can see it in the crowd, the way we played - they're getting better and better. They certainly never deserved to lose the game."