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.
Second-placed Atletico Madrid can close to within six points of leaders Barcelona with a win at home to Celta Vigo later on Friday (2100 GMT).
Unbeaten Barca, who thumped Atletico 4-1 at the Nou Camp last weekend, play at Real Valladolid on Saturday (1700) and can end the year with a remarkable 16 wins in 17 La Liga matches this season.
Third-placed Real Madrid are looking to bounce back from last weekend's 2-2 draw at home to Espanyol when they play on Saturday at Malaga (1900), who are five points behind Jose Mourinho's side in fourth. (Reporting by Iain Rogers, editing by Ed Osmond)