Allardyce, who was Three Lions boss for 67 days two years ago, heaped praise on his former side and believes that they are leading by example in their set pieces.
"You have got to give the management and backroom staff a huge amount of credit because this England team is by far the best in the World Cup on maximising set-plays," he told Sky Sports News.
"They scored two corners against Tunisia, one corner to get them underway today (Sunday) to get them off the mark, they scored a brilliantly-worked free kick.
"Most of the goals have come from set-plays, which is a massive credit to the backroom team. You have also got a penalty and an exceptionally good goal from Jesse Lingard. England are moving forward with a great amount of confidence."
England will play Belgium in their final group-stage match on Thursday with both sides on maximum points after two games.