The Rangers had a four-game winning streak halted and they lost for just the second time in eight outings.
Valverde was without a job for most of the first month of the season before being signed by the Tigers, with whom he had spent the previous three years.
He came on in the ninth to protect a one-run lead and walked Geovany Soto with one out. Ian Kinsler ran for Soto and moved to second on a ground out. Kinsler was stranded at second when Craig Gentry grounded out to end the game, giving Valverde his fourth save.
Four Detroit relievers combined to allow four base runners over 3 1/3 innings, but they salvaged a victory for starter Rick Porcello (2-2).
Porcello allowed a run on five hits with six strikeouts in 5 2/3.
Texas principal bans speaking Spanish, stirs controversy
Ron Burgundy interviews Peyton Manning on SportsCenter