Kervezee, who had just two first-class scalps previously, removed captain Ed Joyce (82), Matt Machan (55) and Luke Wright (21) either side of lunch as the hosts collapsed from 129-1 to 194-5.
Spinner Sachithra Senanayake then took 4-50 as Sussex were bowled out for 290.
Requiring 247 for victory, the visitors reached 47-0 at stumps with first-innings centurion Richard Oliver on 23.
Worcestershire's batsman will be hoping there is no repeat of their second-innings collapse in the defeat against Yorkshire, when they were skittled for just 100, as they search for a first win since their return to Division One of the County Championship.
After Charlie Morris had Luke Wells caught at slip by Alex Gidman in just the second over of the day, Joyce and Machan shared a second-wicket stand of 119 as Sussex looked to be taking control.
The turning point came just before lunch when off-spinner Kervezee trapped Scotland batsman Machan lbw and then had Wright caught at short leg and Joyce snaffled in the slips as he continued his spell after the interval.
Ben Brown (44) and Ashar Zaidi fell either side of the tea break and Sri Lankan Senanayake bamboozled the tail as Sussex lost their last five wickets for just 57 runs.
Worcestershire openers Oliver and captain Daryl Mitchell (16 not out) then batted confidently for 14 overs to leave their side needing 200 more for victory on the final day.
Worcestershire's Alexei Kervezee:  "I must admit it was a surprise when Daryl (Mitchell) asked me to bowl and I was delighted with how it went.
"I have tried to bowl a bit quicker and there was some rough to aim for and I'm pretty chuffed to have got Joyce and Wright out - they are not a bad couple of scalps for my first Championship wickets.
"All the bowlers contributed then our openers did well to get through that spell before the close so we are in a good place."
Sussex all-rounder Luke Wright: "We would have liked to set a few more but we've seen that the odd ball does something out of the ordinary and we've seen throughout the game that wickets do fall in clusters.
"We've still got a great chance because the bowlers always feel they are in the game.
"I got a ball that popped up off a length which I wasn't expecting and it won't get any easier to bat on the final day."

A new report said the gap between the hourly earnings of the two sexes widens after women become mothers.
Their hourly pay rate falls 33% behind men's over a 12-year period, it added.
But in Northern Ireland, there is an established trend of women earning more than men.
In 2010, typical (median) female full-time hourly earnings, excluding overtime, in Northern Ireland moved slightly ahead of male earnings for the first time.
That has continued over the last five years, except for 2012 when typical female and male full-time hourly earnings were the same.
In 2015, typical female full-time hourly earnings, excluding overtime, were 101% of male earnings compared to a ratio of 91% for the UK as a whole.
The pattern of women earning slightly more than men is not uniform across the income distribution.
Men have higher earnings in the 90th percentile (among the top 10% of earners), and at the 40th percentile and below.
The Office for National Statistics (ONS) has suggested that the gap in favour of full-time female workers overall is partly to do with a higher proportion of public sector jobs in Northern Ireland than in the rest of the UK.
There are more women employed in this sector than men and these jobs tend to be higher paid, in general, than in the private sector.
At a wider, all-employee level, there is still a gap in favour of men because there are more full-time male employees.
All of the above data is from the Annual Survey of Hours and Earnings.
Part-time bowler Alexei Kervezee took three wickets for Worcestershire to set up a tense final day against Sussex.
[NEXT_CONCEPT]
Full-time female workers in Northern Ireland are defying the gender pay gap to earn more than their male counterparts, official figures have shown.