Solanke, signed from Bournemouth in a deal worth up to £65 million, has had a slow start to the season and missed two games after making his debut due to an ankle injury.
The striker's 68th-minute goal against Qarabag was his second in as many games following his strike in a league win over Brentford and wrapped up a comfortable win for 10-man Spurs.
"He's getting there," Postecoglou told reporters. "He didn't do pre-season with us and then he has one game and then he's out for two weeks, while the other guys were building up their match fitness.
"Great for him to get another goal but his all-round performance was really strong. When you're down to 10 men, you need your striker to be able to hold up the play and do some hard running."
Postecoglou also praised 23-year-old attacker Brennan Johnson, who has scored in three straight games since Spurs' 1-0 defeat to Arsenal.
"Brennan was really important for us, particularly today going down to 10 men pretty early, we're going to need some attacking outlet," Postecoglou said.
"Brennan's been good at taking up those positions and him and Dom have a good little sort of relationship there in terms of working off each other and he took his goal well."
