The Arsenal striker struck twice to down Vladimir Petkovic's side in their Euro 2016 qualifier, whilst also easing the scrutiny on the under-fire 67-year-old.
Roy Hodgson's men were worth the three points in their toughest qualifying fixture in Group E - which also contains Slovenia, Estonia, Lithuania and San Marino - with Welbeck's close-range finish just before the hour and injury-time strike proving enough to inflict defeat on Vladimir Petkovic in his first game in charge of Switzerland.
Hodgson, who led Switzerland to 1994 World Cup, asked for England to up its performance level after an underwhelming friendly win over Norway last week. And the visitors responded by edging a tight game, although Switzerland had chances to snatch at least a point.
England's breakthrough came in the 58th minute when Welbeck capped a superb team move, and the new Arsenal striker - who came in for the injured Daniel Sturridge - added a second in stoppage time as he latched onto a Rickie Lambert pass and dispatched past Yann Sommer.
England created promising openings in the early stages with Raheem Sterling particularly lively.
The Liverpool teenager orchestrated a four-on-two counterattack after 15 minutes by showing the Switzerland midfield a clean pair of heels, but his pass to Wayne Rooney was mishit and the captain’s tame shot went straight at Sommer.
The visitors wasted another great chance to take the lead just before the half-hour mark.
Welbeck nipped in front of Steve von Bergen before bursting into the area with Sterling and Rooney in support, but his pass failed to pick out the former and Switzerland cleared its lines.
And that wastefulness was almost punished when poor England defending handed Switzerland the best chance of the half.
Jones sloppily gifted possession to Xherdan Shaqiri, whose through ball found Seferovic in the area, but the striker’s shot was well saved by the foot of Joe Hart.
Jones then came agonisingly close to making the breakthrough himself, the center back meeting Rooney's corner with a firm header and forcing a superb save low to his left from Sommer.
England started well after the break too, with Jack Wilshere combining well with debutant Fabian Delph before dragging a left-foot shot narrowly wide of the right post.
Switzerland remained dangerous, though, and Seferovic completed a flowing move with a ferocious half-volley that drew a terrific athletic stop from Hart.
Hart's save proved crucial soon after as Welbeck broke the deadlock.
Rooney collected the ball in the center circle and sent Sterling free down the left, with Welbeck finishing the 19-year-old's inch-perfect delivery from just outside the six-yard box.
Switzerland sought a response and one almost arrived from substitute Josip Drmic.
The forward appeared offside as he collected Shaqiri's throughball before he rounded Hart and shot towards goal, but Cahill recovered to make a superb goalline block.
And Welbeck was on hand late on to seal an important win, as Lambert slid him in and he shot coolly beyond Sommer in the dying stages.