Trippier got the home side off to a flying start as he scored a free-kick after just five minutes, with Ings grabbing his fourth of the season just after half an hour to give Burnley some breathing space.
And despite Alex Cisak being much busier in the Burnley goal after the break and Jason Shackell needing to clear off the line, the home side held on to go into the hat for Wednesday's third round draw.
Burnley got off to a flyer as Ings' dipping 30-yard free-kick deceived Thorsten Stuckmann in the Preston goal after Sam Vokes had been tripped.
Five minutes later the Clarets came close to doubling their advantage when Trippier delivered a deep cross from the right that Vokes headed over at full stretch at the far post.
And the early assault form the home side continued as David Edgar saw his header from Keith Tracey's corner headed off the line.
Finally Preston weathered the storm to gain a foothold, and North End's Stuart Beavon forced a neat save from Cisak at his near post after weaving into the area.
And from the resulting corner, Tom Clarke rose highest to plant a header just wide of goal with Cisak scrambling to cover.
However, the crucial second goal in the 32nd minute stopped the promise of a fightback dead in its tracks as Ings weaved his way through the Preston defence before planting a shot into the bottom corner.
Cisak confidently preserved that lead with another smart save eight minutes before the break, diving to his right to parry away Bailey Wright's header.
North End knocked on the door again straight from the restart as Iain Hume nodded down for Beavon whose flicked header was cleared off the line by Shackell.
And moments later, Cisak was relieved to see Joel Byrom's drive from distance fizz just wide of his left hand post.
In the 55th minute Byrom got his angles right from an arcing 30-yard free kick, but this time Cisak was equal to the dipping ball and tipped it safely over for a corner.
But as the visitors pressed the threat was always there on the break and Ings might have wrapped things up when he tried to walk the ball around Stuckmann on the hour, only to be denied by the German goalkeeper.
Michael Duff volleyed over from a half-cleared corner before North End went for broke, replacing Lee Holmes and Hume with Nicky Wroe and Kevin Davies.
But by now North End's threat had diminished and the Clarets comfortably ran down the clock without any further danger.