Valencia made heavy weather of the victory at their Mestalla stadium but held off the Madrid club reasonably comfortably in the end to climb above them into ninth.
Both have 24 points from 17 matches but Valencia have a slightly superior goal difference.
Valverde was appointed at the beginning of the month to replace Mauricio Pellegrino, sacked after the club's erratic start to the season.
Valencia have won four and lost one in all competitions since but still have a lot of work to do if they are to match the third-placed La Liga finish achieved for the past three seasons.
After Soldado's early strike, Jonas added a second for the home side in the seventh minute and Alvaro Vazquez pulled a goal back for Getafe six minutes later.
Ricardo Costa made it 3-1 on the half hour before an Alexis header gave Getafe a sniff of hope with just over half an hour left that was snuffed out by a stoppage-time goal from substitute Nelson Valdez.