Albion survived a late scare in their opening match of 2016 but ran out 26-24 winners at Priory Lane on the London club’s new artificial pitch.
The victory was the city side’s fifth in a row and moves them up to fifth in the National One table.
“It was a tremendous win for the lads,” said director of rugby Dawe. “To get two away wins like that on the bounce is a big thing.”
Albion had led 18-6 at half-time against Rosslyn Park and on two other occasions were 10 points up, but the home side nearly grabbed a draw at the death.
Replacement Tom Whiteley scored a try with the last play of the game, but former Albion development player Tom Whelan missed the conversion.
“It did go down the wire,” said Dawe. “There were a couple of smoking-good tackles on the line from us, which drove them out wide to score and then there was too much pressure for the kicker really.
“It was not the hardest kick in the world, but it wasn’t a give-me for sure.”
Albion had lost by one point to last season’s runners-up Rosslyn Park at Brickfields in the second match of the season.
“The games show that the two teams are very close on ability and squad strength,” said Dawe.
Albion had the better of the first half on Saturday, but Rosslyn Park dominated territory and possession after half-time.
“In the first half we got a couple of really good turnovers and capitalised,” said Dawe. “We built pressure, finished well and controlled field position.
“I was a little bit upset we did not turn them over in the tackle more in the second half as they were going light and we didn’t get near their line-out.
“But Billy Searle and Marc Koteczky did the team well and those little things are what ultimately wins you the game.”
Saturday was the first time many of Albion’s team had played a match on an artificial pitch.
Dawe said: “A surface like that when dry is one thing, but when it’s wet you do slide in the tackle and gain an extra metre, especially on the power game. Ultimately that is where they got one of their tries.”
Albion will be back in action at home for the first time for a month this coming Saturday when they entertain Hull Ionians.