South African star Kleinveldt smashed a superb 91 from just 88 balls as David Ripley’s men eventually made it to 321 before being bowled out in 86.4 overs.
Saif Zaib was also in the runs, securing a maiden first-class half-century before finishing unbeaten on 65 from 113 deliveries.
Laurie Evans was another man to make it past the half-century mark as the Warwickshire loanee registered an impressive 74 on his debut.
Northants then started brilliantly with the ball, with Ben Sanderson grabbing two wickets in an over.
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Muhammad Azharullah went one better as he claimed three scalps, with Glamorgan limping to 139 for five ahead of the third day.