Off-spinner Raza claimed three wickets apiece in the morning and afternoon sessions to finish with figures of seven for 113.
Timycen Maruma, who last appeared in a Test match for Zimbabwe in 2013, has been named as Kasuza's concussion replacement.
The visitors are still 284 runs behind the hosts who lost their last four wickets in the morning to be bowled out for 406.
The Zimbabwe captain, who won the toss and chose to bat first, made 107 from 137 balls.
Zimbabwe had recovered from the loss of three early wickets to Lakmal to reach lunch at 120 for three.
Zimbabwe, playing their first Test since November 2018, ended the fourth day on 30 without loss and trailing by 127 runs
Mendis battled his way to 80 while former skipper Mathews was far from fluent in reaching 92 not out from 253 balls
Tiripano had earlier made a handy 44 not out batting at number eight as Zimbabwe advanced from their overnight 189 for two to 358 all out.
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