United had won just two of their previous six games across all competitions before Thursday's last-32 first-leg clash in Turin, but they turned on the style against their LaLiga opponents.
Bruno Fernandes scored either side of half-time to take his goal involvement tally in the competition since his debut in February 2018 to 24 (16 goals, eight assists) – more than any other player.
Marcus Rashford and Daniel James then sealed the biggest away victory by an English side against a Spanish opponent in the UEFA Cup/Europa League, surpassing Newcastle United's 3-0 win at Mallorca in 2004.
Solskjaer was thrilled with his side's display, which stretched their unbeaten run away from Old Trafford to nine games, and says there is plenty more to come from them.
"We scored some good goals and played some nice football," he told BT Sport. "We deserved to win but maybe not by four goals. On a night like this when we find spaces we can be a dangerous team.
"We had to defend well. They have some terrific players up front and we earned the right to counter-attack quickly. They opened up and we got the spaces we wanted.
"Every team attacks you, it's about finding the moments to find the space. We are a work in progress, we are learning and some games we have not been able to win lately. We got our X-factor back today."
Manchester United host Newcastle United in the Premier League on Sunday before La Real visit Old Trafford for the second leg next Thursday.