The Seagulls arrived on Teesside having won their last three league games to move within touching distance of the Championship play-offs.
However, they conceded what proved to be the game's only goal in the 22nd minute when Ankergren, under no pressure, spilled a left-wing cross from Joe Bennett and Scott McDonald pounced to convert from close range at the second attempt.
Brighton had hit the woodwork through Kazenga LuaLua prior to McDonald's strike, his fifth of the season, yet they created little else and could have few complaints about the final result.
Middlesbrough, who now lie just five points behind leaders Southampton in fourth, also struggled to display their best form in the final third, but that is unlikely to be of great concern to manager Tony Mowbray after his side held out with relative comfort for only their fourth home league win this term.
Barry Robson returned to Boro's starting line-up after recovering from a thigh injury, while Marouane Zemmama was promoted from the bench to replace Faris Haroun.
Brighton made three changes, with Romain Vincelot, Matt Sparrow and LuaLua coming in for Alan Navarro, Ashley Barnes and Billy Paynter.
The hosts were the first to threaten when Zemmama volleyed wide from the edge of the area in the sixth minute.
A succession of Middlesbrough corners followed, but Craig Noone then fired narrowly wide for Brighton before a free-kick was touched to LuaLua, who struck the outside of Luke Steele's right-hand post from 25 yards.
Boro were gifted the lead soon after as Ankergren failed to deal with a seemingly innocuous cross from left-back Bennett.
Zemmama then limped off to be replaced by Malaury Martin, before a furious Robson went into the book for dissent, the Scot having accused LuaLua of taking a dive.
Ankergren had to be alert to parry away another Bennett cross shortly before half-time, while Martin, who scored a stunning injury-time winner against Bristol City last weekend, fired wide from a free-kick.
Brighton began the second half with renewed energy and Gordon Greer should have done better when heading over from a LuaLua corner.
Following a series of late challenges from players on either side, Matthew Bates then wasted a golden opportunity to double Boro's lead in the 53rd minute. The skipper was left unmarked from Tony McMahon's inswinging free-kick, but volleyed over from 10 yards.
Ankergren was once again guilty of poor handling five minutes later as he fumbled another McMahon set-piece towards Robson, who could not react quickly enough to head home the rebound.
Boro continued to press forward as the hour mark passed, but they were unable to find a second goal and suffered another injury blow when Marvin Emnes was forced off, Bart Ogbeche coming on in his place.
There was more positive news for the Teessiders when Kevin Thomson returned from his hamstring problem as a late substitute for the influential Robson.
Ogbeche had a chance to make things more comfortable for the home fans late on, only to screw his shot wide after fine work from McDonald, who twice forced Ankergren into action during five minutes of stoppage time.