The Red Devils are now seven points behind leaders Manchester City, who have a game in hand, but Solskjaer insists his men will not allow the Citizens to run away with the title.
Manchester United started the game slowly in West Midlands as the visitors took the lead through striker Mbaye Diagne inside the opening two minutes.
The Red Devils pulled parity on the cusp of halftime kind courtesy of Bruno Fernandes but that was not enough to secure the maximum points as they settled for a draw.
"I said we shouldn't be talked about [in the title race]. A compliment for the boys we got into position we are now,” Solskjaer told reporters after the game.
“We won't let them [Manchester City] run away with it - we're playing them soon. We're not giving it away early - no we're not."
The Norwegian coach also reflected on his side’s stalemate against Sam Allardyce’s men.
"It's something we knew would happen from the kick-off - we knew they'd put us under pressure and put in a few throw-ins. They put a cross in and the boy finished well,” he added.
Solskjaer was further asked if there was a foul for the Baggies goal. He said: "As an ex-striker I'd say that was a goal. But in a recent game Harry [Maguire] got on top of a defender and that was no goal. It's just clarity and consistency. I'm not saying that's a foul.
"The goal we scored is the way we put pressure on them. We did get down the sides sometimes but didn't have anybody to tap them in. One of these days we weren't there where we should have been
"We should play better but we gave ourselves a tough start. You have 90 minutes to make amends but we didn't build that momentum until late in the first half.
