Following on on 193-3, Rikki Clarke and Ateeq Javid started the day for Warwickshire who are interested in signing Middlesex's Eoin Morgan and added a further 17 before the latter was caught by Brophy.
Clarke batted on to make 127 not out which included eight fours and three sixes, as Tim Ambrose was out for 16, Stefan Piolet made four, Neil Carter 21, Imran Tahir 41, Boyd Rankin run out for a duck, and Jim Troughton for 17 as Warwicks were all out for 371.
That left Yorkshire needing 199 to win in the final session.
Boyd Rankin had Adam Lyth caught with the score on five but Yorkshire were now in limited overs mode and going for the victory.
Rankin picked up his second wicket when he had McGrath lbw only for Jack Rudolph and Jon Bairstow to start pilling on the runs.
Rudolph made 80 from 91 balls before he was run out by Rankin and Warwicks took a fourth wicket when Brophy was bowled by Imran Tahir.
But Jon Bairstow finished on 64 not out as Yorkshire reached 200-4 in 34.4 overs to win by six wickets.
Tomorrow, Warwickshire travel to Northamptonshire for a Twenty/20 match. Play is due to start at 5.30pm.