The hosts were 170 for three at stumps, with Mendis (60) batting alongside Angelo Mathews (17) at Colombo's R Premadasa Stadium.
Sri Lanka's highest-ever successful run chase was against South Africa in 2006, when they achieved their 352-run target in Colombo.
Zimbabwe skipper Graeme Cremer dented the hosts with his leg-spin, claiming the important wickets of Upul Tharanga (27) and his opposite number Dinesh Chandimal (15).
Opener Dimuth Karunaratne was bowled by left-arm orthodox spinner Sean Williams for 49 as the visitors persisted with an all-spin attack in the 48 overs bowled in the innings so far.
Zimbabwe were earlier bowled out for 377 in the second session, with Sikandar Raza (127) top-scoring for the visitors with his maiden Test century.
Cremer was the last man out for 48 off left-arm spinner Rangana Herath, who returned figures of 6-133 to take his match tally to 11 wickets.
The Pakistan-born Raza steered the team's middle and lower order to help his side accumulate another 318 runs after being reduced to 59-5 on day three.
Zimbabwe are seeking their first-ever Test win against Sri Lanka, who are looking for redemption under new Test captain Chandimal.
He took over from Mathews, who resigned after calling the one-day series loss a "hard pill to swallow".
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