Two days had been lost to the rain and Warwicks who were all out for 154 in the first innings, bowled Essex out for 216.
Rankine took 4-80 with Chris Woakes taking 3-28 whilst Napier top scored for the visitors to Edgbaston with 75 runs.
Warwicks batted again and were 68-2 with Darren Maddy 30 not out when both skippers decided that a result was not going to happen and the match was declared a draw.
Warwickshire's next match is a Pro 40 game at Glamorgan next Friday.