The Scot is looking for the club's second away victory in a row to help start a run up the league table, though he admits it was not easy for his side - even against 10 men
The Scot has again endured a rough start to the season on Merseyside but realises his teams tend to start slowly only to finish strongly in the second half of the campaign.
"It wasn't comfortable. But then I don't think any game is in the Premier League," he told Sky Sports.
"Bolton made it difficult with 10 men and I just wanted to get the boys in at half-time and have a word about how we could play better against the 10.
"Bolton certainly kept working hard when we got 1-0 up, but overall the players have done well over the week and they deserve it."
Moyes' side has now won two away games in a row, and the manager insists he has believed in his team's quality despite it not being reflected in the table thus far this term.
"I've always said we had a good team here, we just needed to start showing it by our results," he stated.
"If you look at our history over the past few years we've been slow starters and then come good and I hope that's it now and we can keep this run going until the end of the season."
Moyes agreed with the early match-changing penalty decision after David Wheater clattered into Diniyar Bilyaletdinov on 20 minutes.
"I thought he was right to send Wheater off," Moyes continued.
"I don't think he's the type of player to hurt someone but he's over-run it and he didn't half give the boy a clattering on his ankle. So for me it was a red card.
"Fortunately the damage wasn't too serious, but as I say I don't think the fella went out to hurt him and if one of my players had done that I would have expected him to be sent off."