Mongia's Leicestershire restricted Lancashire to 194 for eight but was then skittled out for a meagre 125 in the 41st over at Old Trafford.
Asked to bat, Lancashire lost an early wicket but Mal Loye (32), Andrew Crook (42), Stuart Law (42) chipped in with useful knocks to see the side reach 194 for eight.
Mongia, with his gentle left-arm spin, ran through the Lancashire middle order, claiming the important wickets of Crook, Law and then removed Glen Chapple and Dominic Cork to emerge as the pick of the Leicestershire bowlers, claiming four wickets, conceding 25 runs in his seven overs.
Despite restricting the opposition to a chaseable total, Leicestershire was never in the game and three-wicket hauls by the pace duo of Tom Smith and Sajid Mahmood helped Lancashire skittle out their rivals for 125 in 40.4 overs.
With the bat, Mongia scored 22 from 37 balls with five hits to the fence. But neither Mongia, nor John Maunders (38) could make up for the abject display by other batsmen and ended on the losing side.
Zaheer Khan, on the other hand, kicked off his county stint in style, helping Worcestershire edge out Nottinghamshire by two wickets via Duckworth-Lewis system at Trent Bridge after a rain-delayed start.
Inserting the hosts, Worcestershire bowlers, especially Zaheer, used the gloomy overcast conditions to put them on the backfoot and Nottingham were gasping for breath at 70 for six.
Mark Ealham (51) and Chris Cairns (27) then arrested the slide but Zaheer castled Cairns to trigger another batting collapse and then came back to get rid of Graeme Swann (7) and Ryan Sidebottom (5) as Nottinghamshire lost all their wickets to reach 161 in 41.32 overs.
Chasing a target of 162 in the match, that was reduced to a 43-overs-a-side affair, Worcestershire reached 165 for eight in 42.3 overs to win the match with contributions from Stephen Moore (25), Steven Davies (33), Ben Smith (41) and Graeme Hick (22).