The young striker opened the scoring after 20 minutes but Leyton hit back with two goals in three minutes to lead 2-1 at the break.
Fulop's moment came 10 minutes into the second half after we were penalised for handball before Malcolm grabbed his second to level it at 2-2 with 20 minutes left. That's how it stayed.
Coach Clive Allen takes up the story: "We played really well in the first 20 minutes and managed to go 1-0 up with a well-worked goal. Phil Ifil linked up with Mark Yeates and Michael was in one-on-one and finished well.
"But we went to sleep after that goal and conceded two goals in three minutes.
"The first came straight from kick-off and a long throw that we didn't clear. The second came after we lost possession, their striker ran at our central defence and to be fair, finished well.
"So from scoring after such a good start we've gone in at half-time 2-1 down. The lads were disappointed, realised what that had done and wanted to put it right.
"I thought we went about it in the right way in the second half but Leyton still had their tails up and could have made it 3-1 when they were awarded what I thought was a harsh penalty for handball. However, Marton made a great save down to his right.
"We finished the stronger side and Michael scored a very good second goal, good movement again, John Kyriacou did well down the left, Jamie O'Hara played in Michael and he took it well.
"Overall there were good and bad points but I wasn't too disappointed."
Spurs XI: Fulop, Ifil, Kyriacou, McKenna, O'Donoghue, Welch, Price (Barcham), O'Hara, Malcolm, Yeates, Daniels (Seanla). Unused subs: Eyre, Mascarenhas, Lewis.
Michael Malcolm netted twice and Marton Fulop saved a penalty as a young Spurs XI drew against Leyton FC in Tuesday night's friendly.