Instead the visitors were made to work all the way by a dogged Dragons outfit, but despite their tenacity Beck's second-half effort resigned them to a sixth defeat in seven RaboDirect PRO12 games.
The Ospreys were ahead in their first attack through Walker who finished expertly in the left corner after a sweeping move.
The wing was tackled by Will Harries but somehow kept his feet in the air as he was being dragged to touch. Dan Biggar, who finished the game with four points, slotted the conversion from the touchline for a 7-0 lead.
Moments later Tom Prydie missed the chance to reduce the arrears for the Dragons with a penalty, but on 12 minutes he made no mistake after the Ospreys captain Kahn Fotuali'i was penalised ten yards for dissent.
Despite their initial bright start the Ospreys were struggling, with handling errors gifting Newport territory.
However, Steve Tandy's troops stood up to wave after wave of attacks from the hosts and they escaped midway through the first half with their lead intact.
A ten-minute lull of kicking between the side's backs saw the Dragons charge halt somewhat, but Biggar came up short with a penalty six minutes before the break.
Two minutes after the restart Walker came within a whisker of notching his and Ospreys' second try but prop Phil Price palmed the ball into touch before he could ground it.
The Ospreys were beginning to up the pressure and seven minutes later Biggar attempted a clever kick that Prydie had to collect beyond his try-line, otherwise Ross Jones would have been in to score in the right corner.
But their second-half dominance eventually told when replacement Beck went over with his first meaningful contribution of the match on 62 minutes.
The centre, on for Jones just four minutes previously, latched onto a Biggar charge down to find his way under the posts. Biggar's conversion made it 14-3 to the visitors.
Despite a renewed effort from Newport to gain a losing bonus point, they found Ospreys' defence too determined to break and the Liberty Stadium outfit held on to move onto the coattails of fourth-placed Glasgow Warriors in the RaboDirect PRO12 table.
|Newport Gwent Dragons Score Card|
|Ospreys Score Card|