Roddick was never at his best against the gutsy Beck on Court One, but the American eighth seed's superior grasscourt skills eventually prevailed to set up a clash with either Jaroslav Pospisil or Victor Hanescu.
"The second set tie-break was big for me because there's a big difference between two sets up and one-set all," Roddick said.
"He had a pretty smart game-plan and was going for his shots a lot, but I felt like I was playing better than him so I didn't want to let him back in.
"I'm pretty happy with where my game is right now. I feel healthy and I know what it takes to go deep at this tournament."
Finally feeling close to 100 percent after a shoulder injury forced him to miss the French Open, Roddick reached the semi-finals of the Wimbledon warm-up event at Queen's two weeks ago.
But the 28-year-old suffered one of the most comprehensive defeats of his career in the semis as he won just four games against Andy Murray.
Roddick, beaten by Roger Federer in all of his Wimbledon finals, had insisted that humbling loss wouldn't damage his confidence at the All England Club and he proved the point in a first round tie originally postponed on Monday due to rain.
Beck, ranked 156th, has never been past the second round of any Grand Slam and, although his lusty hitting caught Roddick off-guard at times, the result was rarely in doubt.
Roddick's formidable serve was working well enough and he sealed the first set with a break at 5-4.
The second set followed the same patten, with Roddick doing all the pressing and Beck fighting to stave off three break points at 3-4.
That only delayed the inevitable as Roddick took the set when his blistering return of serve forced Beck to net a volley.
Roddick, surprisingly beaten by Lu Yen-Hsun in the fourth round last year, quickly moved a break ahead in the third set and it wasn't long before he was serving out a comfortable victory.
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