David Mooney took his goals tally into double figures for the season when he headed home in the 90th minute to win the Johnstone's Paint Trophy southern area semi-final.
The east Londoners were forced to play the final 20 minutes with 10 men after midfielder Anthony Griffith was sent off.
Slade said: "I'm really proud of my players. We've dug in and we are still in two cup competitions. As a club, as a fan-base, as a manager and as a chairman we've all got to be really chuffed and happy with that.
"It was a hard game. Yeovil are a very direct side who play with tempo and we had to deal with that side of it and match it.
"We did that and although it wasn't a great spectacle in the first half we managed to grind them down and we created a lot more in the second period.
"We probably could have had a couple of goals in the second period if the truth be known and I have to say I thought we were denied a stonewall penalty on Kevin Lisbie.
"But, in the end, we got what we deserved because I thought we were the best team overall when we were paying with 11 men and also when we went down to 10 men.
"It was a fantastic header from David Mooney, whenever the ball goes in the box now Moons is on the move and he's a threat. He scored a great header at Hull last Saturday and he's done the same again."