Pep Guardiola's side dominated the vast majority of this last-16 first leg and the City manager might have been frustrated by the failure to turn that into a more commanding lead until Jesus prodded in after 65 minutes.
Silva and Joao Cancelo created that goal after the Portugal duo had also combined for the 29th-minute opener, with the full-back's dipping delivery from the left for his diminutive compatriot to head home a truly sublime moment.
It brings a familiar hurdle into sight for the Premier League leaders, who have fallen at the quarter-final stage in each of the past three seasons.
Gladbach soon found themselves hemmed in by City's slick passing and astute pressing, with last-ditch challenges needed to thwart Jesus and Raheem Sterling.
Guardiola's men were content to bide their time and the opener arrived just before the half hour, with Silva nodding in Cancelo's sumptuous cross from deep at the back post.
Ramy Bensebaini sloppily gave the ball to Jesus in the 53rd minute but he hesitated and let Nico Elvedi get back to make a goal-saving challenge.
A rare moment of encouragement arrived for Gladbach as the hour approached, with Denis Zakaria powering past Rodri but overhitting a pass into the City box that Ederson swept up easily.
The Brazil goalkeeper was scampering in the 63rd minute when Alassane Plea's audacious flick bounced just beyond the far post.
Those minor scares roused City and Jesus showed sharp poaching instincts when Silva again played the unlikely role of back-post target man to tee him up.
When Phil Foden thrashed over from Ilkay Gundogan's 78th-minute knockdown, it was no surprise that Cancelo's chipped ball into the box crafted the opening.
A careless pass in stoppage time from Rodri gave Gladbach substitute Hannes Wolf a chance to reduce the deficit but Ederson prevailed in their one-on-one.
What does it mean? Savvy City look primed to push for glory
As Gladbach failed to register a single touch inside their opponents' box during the first half at Budapest's Puskas Arena, it was impossible to imagine this City getting involved in the sort of Champions League firefights that have proved their downfall – ties finishing 6-6 and 4-4 on aggregate against Monaco and Tottenham, only to depart on away goals, springing instantly to mind.
Such total control means Guardiola's team are unlikely to quicken the pulse as they did on those occasions but it makes the one major honor to have eluded the Catalan in Manchester a far more likely prospect.
City host in-form West Ham in the Premier League on Saturday, when Gladbach travel to title-chasing RB Leipzig in the Bundesliga.