Boro flew out of the blocks and would have been 1-0 up almost immediately had Jelle Vossen's header that appeared to go over the line been given, but referee Mick Russell disallowed the goal and waved play on.
"From the first minute we were right at it," said Adam. "We got straight in their face and that was our game plan - to press high from the front because Millwall were a team lacking in confidence and weren't going to have 75% possession, so we knew to get at them.
"We had to win second balls which we did. Grant (Leadbitter) and Adam (Clayton) picked them up, we hit them on the counter and at times we were deadly."
"It probably could've been double figures and at half-time it could have been five or six, that's what the manager said.
"He (Aitor Karanka) told us at half-time to go out in the second half and play as if it was 0-0 again and win the second half.
"Their goal came, but there was a good chance of that happening because they had nothing to lose and were throwing men forward but we replied instantly and that's what we're all about this season.
"Winning 5-1 away to Millwall has put a big message out to the other teams."
Boro were looking for a win last Saturday after the team had drawn the last three games and slipped to fifth in the Championship.
And Reach said the win against the Lions was great for the team's confidence ahead of the visit of league leaders Derby on Saturday.
"That win is going to bring a lot of confidence," said Adam. "You know we've drawn the last few games which is disappointing, but we haven't lost and if we keep putting in this effort and the team morale stays up then those draws will turn into wins.
"We're climbing that table and I think the most important thing about Saturday's win was the way we played. We didn't rest on our laurels - we kept going and could've got six or more.
"We've sent out a big message by going to supposedly a hard place like the Den and turn them (Millwall) over like we did.
"I think it's set us in great stead to play Derby on Saturday which is the big one.
"We'll get a couple of days now to rest then we've got another important week to prepare and make sure we're at 100% and I'm sure whatever team goes out there on Saturday will give their all to try and hopefully emulate last Saturday's result."
Tickets for Saturday's game against Derby are available online here, by calling 0844 499 1234, or by visiting the Ticket Office at the Riverside Stadium.