United's Ryan Flynn struck a post with a second-half shot while home keeper George Long came to his side's rescue with a fine double save to keep out Romain Vincelot's shot and David Mooney's follow-up.
Orient striker Shaun Batt was also thwarted by Long early on after finding himself through on goal.
He said: "It was a tight game. We came with a certain game plan and even though we had just one striker, we still wanted to create chances and we did that.
"We were very much 4-3-3 when we were going forward and I thought we created a few really good openings.
"Shaun Batt had a great chance in the first period and then a shot from Vincelot which was followed up by Mooney. They were probably our best two chances and we maybe should have got ourselves a goal from that situation.
"We weren't at our best, for obvious reasons, against Crawley on Saturday but we got the response that I wanted and that's all I can ask for. We need to have a game plan and know what we're doing and individuals did their jobs well.
"We felt that we could create openings with the pace that we have on the right-hand side and the pace of Batt.
"It is pleasing when you come to places like Sheffield and Swindon and see that we have a group that's capable of getting a result. I wish we could play the top six teams every week, then I think we might be all right!"