Kartik picked up the wicket of Warwickshire's top scorer Jamie Troughton (57) as Lancashire moved into the safety zone.
The Indian spinner conceded 49 runs in his eight overs after Warwickshire made 208 for seven in their 40 overs following their decision to bat first.
Dominic Cork was the wrecker-in-chief for Lancashire as he took three wickets, conceding just 24 runs in his seven overs.
Troughton and Nick Knight (52) held the visitors' innings together with their gritty display but their effort yielded no results as Nathan Astle top scored with 71 for the Lancs' to take the hosts to victory.
Mal Loye combined well with Astle and made 46 as Lancs romped home with eight balls to spare.