Dylan Stephenson's first-half hat-trick and a well-taken goal from England under-18 international Joe White saw the young Magpies progress after an impressive performance at Whitley Park, with head coach Steve Bruce watching on.
And Winskill - a former Newcastle youth team player who represented United in the competition in the early 1990s before embarking on a career in coaching - was full of praise for his teenage charges, captained by recent Premier League debutant Elliot Anderson.
"I'm delighted for the lads, and the staff," UEFA Pro Licence holder Winskill told nufc.co.uk. "Good days like today, where you work hard all week, cover every little angle and anything you think's going to be important for the game - it's really good when the lads can go and execute exactly what you want to do.
"The manager was there, and all the Academy staff, and it's great for them to see the lads performing at that level. So I'm delighted for them - they've worked really hard all week and it all came together today."
United needed penalties to get past Huddersfield Town in the third round after a hard-fought 1-1 draw in December, before the competition was temporarily suspended in January because of coronavirus concerns, and Winskill admitted that Saturday's display was a big improvement on that performance at the John Smith's Stadium.
He said: "I think we showed the same level of commitment and intensity, which I was really pleased about and I knew we had to do that, because of the way Leeds play. We had to match that - but it was really important that we stamped our own authority on the game, and the lads with the ball today were excellent.
"They played some really good football, created loads of chances and managed to score four which, in any game, is a great achievement."
Stephenson gave Newcastle a 13th-minute lead, slotting home from close range, and the Magpies never looked back, going into the break with a 3-1 advantage before White wrapped up the scoring shortly after the hour-mark.
And Winskill admitted: "I think it's important for any team, at any level, to get the first goal. Confidence rises, people start to relax into the game more and the nerves disappear quicker.
"So it was really important that we started well. We said that during the week - to get the first goal would really help us and to see that one go in and see the lads grow from it was exactly how we wanted it to be."
Blyth-born Stephenson has scored four goals in the Under-18 Premier League this season - one which has also seen him become increasingly involved at under-23 level - but walked away with the matchball on Saturday and Winskill said: "I'm thrilled for Dylan because he is such an honest, hard-working lad.
"He's the type of player any team would like to have in their XI, who's going to really put a shift in for you. Every game and training session, the physical stats come through and he's always top of the charts but we had a chat during the week about the need for him to be consistent with his quality, and he spoke to me about how he thinks people will judge him on goals.
"He hadn't scored as many goals as he wanted to thus far, and to see him score three today was the icing on the cake after that conversation, so I was chuffed to bits for him."
Newcastle face another home tie - potentially this time at St. James' Park - in the fifth round, with Watford the visitors after they beat Lincoln City after extra time last month. The game must be played no later than Saturday, April 17th and Winskill revealed that preparations have already started for that match.
"We've already had a look at Watford - not wanting to pre-empt anything, but we knew the game was going to come around quickly so we wanted to try and get ahead of the situation if we could," he explained.
"I have to say a big congratulations to the staff - the coaching staff, the medical staff, the sports science staff and the performance analysts who work tirelessly all week, every week. It's really nice for them to see the rewards of all of the work that they've put in as well.
"So we'll enjoy tonight, we'll have a day off tomorrow, and then the staff and the players will be back on it again ready to put in place a plan to repeat what we've done today in the next round."
