The home side had failed to make the most of their chances during the opening half but the contest swung decisively in their favour during a lively opening to the second 45 minutes in Friday's Group G European Championship qualifier.
Switzerland were reduced to 10 men after 54 minutes when former Tottenham full-back Reto Ziegler was shown a straight red for having his studs raised in a tackle on Chris Gunter, and a foul on the same player by Timm Klose six minutes later allowed Ramsey to slot his fifth international goal from the penalty spot.
Bale then made the win safe as he raced on to a Steve Morison pass to beat Diego Benaglio for a first Wales goal since the sides met in Basle a year ago.
Wales boss Gary Speed had handed Swansea midfielder Joe Allen a first international start at his home ground with Craig Bellamy returning from suspension.
Allen's inclusion, and that of club mates Ashley Williams and Neil Taylor, gave Swansea three players in a Wales starting XI for the first time since 1982.
Switzerland gave a debut to Basle midfielder Fabian Frei with Klose being partnered in defence by Steve von Bergen after Fulham's Philippe Senderos was a late withdrawal.
And Cesena defender Von Bergen's first contribution was to earn a yellow card for a poor tackle on Morison.
Klose then gave away a needless corner and it almost proved costly, but Darcy Blake wasted a free header after being picked out by Bellamy.
Despite that chance the visitors, with Xherdan Shaqiri and Granit Xhaka looking lively, had much the better of the opening exchanges, looking assured in possession and starving Wales of the ball.
But the hosts looked a threat on the break with the pace of Bale and Bellamy proving a menace.
Bellamy was off-target with two efforts from the edge of the box before a poor first touch meant Ramsey failed to pick out the Liverpool man in acres of space in the Swiss box.
Wales' best chance yet arrived after 37 minutes as Bale's ball into the box picked out Morison, who expertly beat Klose but saw his shot diverted over the crossbar by Benaglio.
Switzerland responded with their best effort as skipper Gokhan Inler's low drive flashed just wide of the post with an unsighted Wayne Hennessey stranded.
Frei then saw his on-target effort deflected wide as Hitzfeld's side finished the half strongly.
The Swiss thought they had grabbed the opener just five minutes after the restart as Shaqiri's free-kick was bundled in by Eren Derdiyok, but the Bayer Leverkusen forward was ruled to have been offside.
It was then the hosts' turn to feel hard done by as Bale appeared to be felled by Klose as he raced into the box, but the appeals were waved away by referee Bjorn Kuipers.
The Dutch official was again the centre of attention in the 54th minute as he showed Ziegler a straight red card after the left-back had gone into a challenge on Gunter with his stud's showing.
Wales soon had another penalty shout turned away from the resulting free-kick as Blake appeared to be bundled over as he battled to reach Bale's inswinging delivery.
But it proved to be third time lucky on the hour mark as Klose's clumsy challenge on Gunter saw Kuipers point to the spot and Ramsey's low effort could not be kept out by Benaglio.
And victory was sealed in the 71st minute as Morison passed between two flat-footed defenders to find Bale in space. The winger raced in on goal and made no mistake as he shot left-footed past Benaglio.