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."
Part-time bowler Alexei Kervezee took three wickets for Worcestershire to set up a tense final day against Sussex.