Richarlison has now scored eight goals this season, with seven in the Premier League
The midfielder has yet to start a game under Lampard, but came off the bench after 58 minutes to help turn the game Everton's way and the 26-year-old played a part in Richarlison's stoppage-time equaliser.
The draw lifted Everton four points clear of 18th-placed Burnley, who host Southampton on Thursday.
Everton had suffered a nightmare start to the game when Harvey Barnes turned in James Maddison's cross early on.
But after substitutes Alli and Salomon Rondon helped the hosts get a foothold in the game, the pair had a hand in Richarlison's eighth goal of the season, which earned a crucial point for Everton before matches against Liverpool and Chelsea.
Lampard said Alli had deserved his chance in the side after being "absolutely spot on" in training over the last two weeks.
"It's not easy to demand from players that are not playing so much to come in and show your manager you deserve a chance but he's done that," Lampard added.
"I was really pleased with [the part he played in the goal] and hopefully that can be a big starting point for him.
"I don't think the story was ever going to be simple with Dele. We all know the situation when it comes to how the last two or three seasons have looked and that high, high level of player that he reached for a big period of time at Tottenham.
"So I don't think it was a case of things being simple and sometimes you have to work and fight through things and we're in that process with him."
The draw denied Leicester manager Brendan Rodgers a first win at Goodison Park, and it wasted a good start for the visitors, who also squandered several good chances towards the end via Maddison.
The Foxes have already reached a Europa Conference League semi-final this season, but they now sit 11 points behind seventh-placed West Ham, who have played two more games.
Everton could not turn successive home victories into a third, but after a poor first half, this will feel as good as a win.
Without Dominic Calvert-Lewin, who suffered a training-ground injury that could keep him out for two weeks, their front three of Richarlison, Demarai Gray and Anthony Gordon struggled to link together.
Gordon's passing and set-piece delivery was particularly woeful, but once Rondon came on to provide a focal point to the attack, and Alli offered some assistance to the impressive Alex Iwobi, the game began to take a different path.
The early goal and dominance by Leicester led to a subdued atmosphere in the first half where despite Richarlison's miscued shot, the home fans were not lifted off their seats until just before the break when Gray fired inches wide from a clever corner.
Leicester had already gone close when Seamus Coleman tackled Kiernan Dewsbury-Hall before the Foxes took the lead.
With Rodgers making seven changes to the side that lost at Newcastle on Sunday, the visitors were able to cut through the Everton defence easily. Dewsbury-Hall and Maddison both caused problems for the Everton defence, who had Yerry Mina back for the first time since February following a hamstring injury.
By contrast, Fabian Delph and Allan were ponderous in midfield with only Iwobi providing the thrust that Everton and their fans craved. But that soon changed in the second half as Alli came on to replace Allan.
Unfortunately for the hosts, Gordon's passing was still off, and Richarlison, who had scored only three goals since the turn of the year, wasted another chance after good work by Rondon and a free header from a corner.
As Everton pressed for an equaliser, gaps opened up for Leicester to exploit and Maddison went close three times.
That ultimately proved costly as Richarlison and his team's persistence earned a crucial point to keep Lampard's side on the right side of the drop zone, regardless of Thursday's results.
-- Courtesy of BBC Sport