At New Road, the home side resumed on 206 but could not stop Tahir and Boyd Rankine mopping up the tailenders.
Tahir finished with figures of 5-67 and the Ireland international recorded figures of 4-56 as Worcestershire were bowled out for 244.
This left Warwicks needing 47 runs to win with most of the day to get them. Neil Carter was out for four but Ian Westwood (18 not out) and Jon Trott (16 not out) along with nine extras ensured that Warwicks picked up their first County Championship win of the season.
Warwickshire's next County Championship match is on Wednesday when Somerset are the visitors to Edgbaston.
Tomorrow, Warwicks Second X1 take on the MCC Young Players at Coventry & North Warwicks ground in Binley Road. Play is scheduled to start at 11am.