Goalkeeper Mike Green kept Orient out until the 87th minute when David Mooney nodded home before Dean Cox pounced to make it two.
Orient now face Alfreton in the second round, and Slade said: "It was all about getting through in the FA Cup and we know how difficult it can be.
"In the end, we got the goals. We wanted them earlier, but to be fair their goalkeeper has pulled off some fantastic saves.
"Credit to Gloucester because they gave every last ounce, but we wore them down in the end, got on top and eventually got the goals.
"The first 10 minutes were Gloucester's best 10 minutes and we didn't cope with that very well but we got a grip in the middle of the park in the second half."