Opting to bat first, Leicestershire were all out for 251 and Mongia's contribution with the bat was 20.
Ali Brown (131 not out) and Mark Butcher (105) led the Surrey charge even though Brown had a life when he was dropped on two in the slips off Mongia.
Mongia first trapped Mark Ramprakash (73) and eventually induced Butcher to edge him to slip and perish.
The left-arm spinner's figures in the match read 26-6-73-2.
In another match, Zaheer Khan's Worcestershire scored 618 in their first innings and then had Somerset at 199 for three.
Zaheer, who was not out on one, castled Matthew Wood for eight to earn the breakthourgh for his side.
Zaheer conceded 50 runs in his 15 overs, which included a maiden, to claim his lone wicket so far.