Rooney nodded home the only goal against co-hosts Ukraine in Donetsk three minutes after the break on Monday night.
The Manchester United forward, back after completing his two-game suspension, had earlier missed a golden opportunity but did not let that affect him and has now scored 29 goals for England - just one behind Nat Lofthouse, Tom Finney and Alan Shearer, who are joint fifth on the all-time list.
"He's got character, hasn't he?" Hodgson said of his talisman.
"He showed them in abundance tonight. He and (Danny) Welbeck worked extremely hard."