Wasps and Munster find themselves in this seasons' "Pool of Death" alongside French challengers Clermont Auvergne and Welsh side Llanelli Scarlets, and a home defeat on the opening weekend would have left the Londoners with a mountain to climb.
But Skivington's touchdown midway through the second period gave Wasps a one-point lead that they held onto despite a fierce finale.
Skipper Dallaglio said: "We won't be getting ahead of ourselves. This is a very tough group.
"It is just about winning and scrapping for everything you can. Invariably, all of the groups will go down to the last couple of weeks.
"At this stage, it is a case of not worrying too much about other teams. This was a great start to the competition, but there are other battles ahead."
Wasps now head for Llanelli next Saturday, a venue where they have already lost twice in the Heineken Cup, and Dallaglio added: "We've got a big challenge. It is going to be a very tough game in front of a full house in a hostile environment."
Wasps trailed 13-3 after the opening quarter, and then 23-13 early in the second period in front of a club record Heineken Cup crowd of 21,500.
But they produced two stirring fight backs, underlining that it is going to take a good team to wrench the trophy from their grasp.
Wasps rugby director Ian McGeechan said: "To come back twice from those deficits was a big call from the team, but we felt we probably blew 15 or 20 points in the first half."
And McGeechan also paid tribute to young England prospect Cipriani, who controlled things well from fly-half and looked assured in the heat of battle.
McGeechan added: "I was pleased with him. His goal kicking was crucial, and with someone as accurate as Ronan O'Gara on the other team, he had to measure up tonight because he was up against a top-class performer."
O'Gara though, struggled to hide his frustration after Munster came unstuck in a game that saw him boot 13 points, while centre Rua Tipoki and full back Shaun Payne scored tries.
Ireland star O'Gara said: "It was a game I felt we left behind.
"At this level of competition small things matter, under 10 points clear, if we had kicked on to 13 or 15 points in front I felt it was game over.
"It could have been a big, big win for us, and I don't think many people would have been complaining if we had won."
Man of the match Alan Quinlan added: "It was definitely one that got away. We are really disappointed, but there is a long way to go in this competition and we will certainly learn from today."