The young Magpies led inside four minutes of their Northern Group F clash at the One Call Stadium following Cathal Heffernan's close-range finish before goals from Max Dickov and Finn Flanagan looked to be eliminating the visitors with a game to spare.
Ollie Taylor's stoppage-time own goal, turning the ball home from Charlie McArthur's cross-cum shot, salvaged a point for United's second-string before gaining a bonus point after defeating the Stags 5-3 on penalties following successful spot-kicks from Sean Neave, Sam Alabi, Joe Brayson, Seung-soo Park and McArthur.
"We know what types of games these are going to be like before we come into them," Lawrence told newcastleunited.com. "From previous games we've played to tonight, we want to see the learning and for them to manage those moments in these types of matches, showing progress from the last game we played, and I thought we did that.
"If you think about the Huddersfield game [a 6-2 loss in their opening Vertu Trophy Northern Group F fixture], we were two goals down after 12 minutes, conceding from a corner and penalty, so to then manage those first parts was really good.
"It was also really tough as Mansfield played the game properly, were man for man all over the pitch and very aggressive. I thought in the second-half, when the game and spaces opened up, we looked more exciting going forward and could impose our style on the game even more.
"Credit to the boys after conceding the second goal as we found a way to stay in the game after that and ended the game strongly and, on the whole, deserved the equaliser.
"On the penalties, it's great to see those players under pressure and having the composure in those moments, executing the technique they want. To score five from five was excellent."
Lawrence commended his side's bright display in the final stages of the contest and believed they warranted a positive result against Mansfield, who currently occupy 12th spot in the English third tier.
The Bermondsey-born coach, who joined Newcastle's Academy last month following his departure from Manchester United, where he was Lead Coach for the club's Under-18s side, was pleased with the player's ability to overcome the difficult situations to ensure a valuable two points.
"In any football match, it's very rare you are going to dominate for 95 minutes," he added. "You will have those spells where you will be under pressure and you have to find a way to get some momentum back in the game.
"You always felt the goal could come at the end and it's just whether you have that bit of quality in that last moment, which we did. It was a lovely pass from Seung-soo to get Charlie in down the left-hand side who managed to squeeze in the finish. It was a really nice way for the group to end the game and I thought they deserved that.
"If you look at the penalties, they weren't just scored but they were clean finishes, with players looking really confident when stepping up. It's fine margins in penalty shoot-outs but the bigger picture is that it's good for our players to experience that.
"We go into the last game [against Harrogate Town] giving ourselves a chance of progressing in the competition, which is ultimately what we want to do."
