Neil Doak's men currently top the Pool C table ahead of their round 2 visit to Moseley this Friday (kick-off 7.30pm).
Billesley Common, the venue for the game, witnessed a high-scoring clash between Moseley and London Welsh last week which the Londoners won on a 38-27 scoreline.
Commenting on his side's winning start to the British & Irish Cup, Ulster Ravens out-half Niall O'Connor said: "We were delighted with the win, that was the main thing.
"I think the first half was pretty exceptional, we put some great phases together - precision and accuracy was spot on - and we scored some good tries as a result.
"I suppose you get four tries in the bag, the temptation is to spread the ball wide to entertain the crowd.
"That is possibly what happened in the second half, it went that way and it turned into a bit of freestyle rugby at times, and we couldn't add any further tries."
While the second half was a disappointment for the Ravens, their run of four tries in the opening 33 minutes against Bedford shows the quality in the Ulster squad and O'Connor explained that the British & Irish Cup is an exciting addition to the calendar for the players.
"Definitely, I noticed that with the build-up to the Bedford game, there was a different feeling to any another other Ravens game because it was the cup.
"It added that bit more intensity to it. In fact the build-up to it felt like a Magners League game for me and for a few other boys. Obviously there was a bit of extra pressure being a cup game, but it was great to get a good result."
Moseley scored three second half tries against London Welsh and will be a tough nut to crack on their home patch.
Former Ulster scrum half Doak will be missing a few players due to the Ireland 'A' v Argentina Jaguares game, which is taking place in Tallaght on Friday.
The Ravens boss is mindful of how crucial these away games could prove to be when the knockout stages come into view in the spring.
"It will be tough to go to Moseley this week. Bedford played them recently and came out on top, but it will be tough on the road," said Doak.
"We are probably going to make a few changes and have a look at a few other boys, and we are going to have a few forced changes as Ireland 'A' are also playing this Friday.
"We just have to knuckle down this week, and in the first few days try to embed a few bits and pieces together for the new guys coming in.
"We have these opportunities with this new cup competition and we are looking forward to another tough game in Moseley on Friday night."