Roy Hodgson's men romped to victory over the minnows in World Cup qualifying in Serravalle on Friday night, with Oxlade-Chamberlain among the scorers.
The Arsenal teenager hailed England's clinical display, insisting it was crucial they kept the pressure on to see out the win after going 5-0 up by half-time.
"We knew it could be a bit frustrating - any team that puts a lot of men behind the ball on the edge of the box can frustrate you," he told Sky Sports.
"Luckily enough we got a breakthrough with the own goal and the boys kept plugging away and we managed to get five before half-time, which was brilliant.
"We wanted that to happen so we could, not take our foot off the gas, but know that we had the result won and see the game out.
"We knew we wanted to come here and get the job done and the more goals we could score, the better for us.
"I think we were spot on from the start tonight, no-one showed any complacency. We showed the opposition real respect throughout the team, not just the boys up front but the boys at the back as well."
Montenegro also took three points to stay top of qualifying Group H as they triumphed 1-0 in Moldova and Oxlade-Chamberlain is expecting a tough encounter when they meet on Tuesday.
"We can't control their results when we aren't playing them," he added. "We just knew we had to get our result.
"They got three points so it makes that game a bit more important when we go there on Tuesday.
"It's going to be a tough game. That one will be very important. We will be approaching it in right way. It's going to be a tough atmosphere and environment with their hostile crowd, but the away support will really help."