Andrew Johnson fired home the first after 66 minutes before linking well with James McFadden who doubled the lead with a cool finish 18 minutes from time.
But despite moving to fifth with the result, Moyes does not want to overplay the significance of making the Champions League - something he has already achieved with the Toffees in 2005.
"When we did it before everybody told us it was a fluke, so we'll have to see if we can change that," Moyes said.
"But the competition above us is very high. We're not kidding ourselves, finishing in the top 10 of the Premier League this season could be a big job, nevermind the top four."
The last time the Toffees reached such heights they beat city rivals Liverpool to fourth but Moyes insists knocking the Reds out of the Champions League - a fate they avoided last time as European champions - is not a motivation.
"We'd be happy to edge anybody out - it doesn't matter if it's Liverpool or anyone else.
"But the boys are playing well and they've played well all over Christmas so this result was very important."
The Scot was also delighted by the way his responded to a host of absences - including midfield playmakers Tim Cahill (knee) and Mikel Arteta, who served the first of a three-match ban against Boro.
"We've no Cahill, no (Joseph) Yobo or (Leon) Osman and Thomas Gravesen is injured.
"Coming here without all those players shows we've not been too affected by the last game (a 4-1 defeat to Arsenal).
"We came here with just about no midfield. Our only recognised midfield players were Lee Carsley and Steven Pienaar.
"It shows the strength in depth and what we have got here. The players who came in were fantastic."
Boro boss Gareth Southgate admitted the psychological blow of having shots cleared off the line twice before going behind - Stewart Downing was denied by Phil Jagielka and Tuncay Sanli by Joleon Lescott - took its effect on his ailing side.
"As is often the case with games at the end of the Christmas period, the first goal has a massive influence," said Southgate.
"If we'd have got it, and we had two cleared off the line, it would have been a huge lift, an extra boost.
"But they got it and they energy drained away.
"We weren't quite at our best but I can't fault what they've given.
"There wasn't the freshness to win the game. They're out on their feet and they must be feeling like every time they take a step forward a door slams in their face."