Ashley Cole has a record seven FA Cup winners medals and now he can also say that he has scored in the competition, but he will not be contesting this season’s quarter-finals, as his landmark goal here was not enough to save Derby from elimination. The 38-year-old full-back scored after being introduced as a half-time substitute but Brighton clung on and advanced thanks to first-half goals by Anthony Knockaert and Jürgen Locadia.

“It got a bit nervy at the end,” said Chris Hughton, whose side had been comfortably superior to the disappointingly limp visitors for most of the match. The Derby manager, Frank Lampard, said he was annoyed with his team’s first-half display but proud of their turnaround in the second.

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By the dying minutes they resembled the side who had produced cup upsets against Manchester United and Southampton this season, but by then it was too late.

So the FA Cup continues to provide solace to Brighton, whose run of one win from 11 league matches has taken them into relegation danger. “I’m hoping some of this positive momentum [from the FA Cup] can carry over into the league and we can pick up more points,” said Hughton, who again omitted most of his regular starters. Brighton were still too good for Derby.

The first half-hour was a gruelling spectacle, with neither team showing much by way of ingenuity or even ambition. Brighton had more of the ball but did little with it. Derby defended deep and were entirely useless in possession. The only early chances to speak of came from a pair of free-kicks, Knockaert and Yves Bissouma bringing humdrum saves from Kelle Roos. But then the Premier League side took charge with goals of uplifting quality.

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Bissouma, whose clever bursts from midfield were one of the highlights of the home team’s play, created the first by playing a slinky one-two with Alireza Jahanbakhsh before picking out Knockaert with a low pass from the right. Knockaert gave the move the finish it deserved, guiding a side-footed shot into the bottom corner from 12 yards.

Derby barely whimpered in response. Martyn Waghorn was presented with a shooting chance following a poor defensive header by Shane Duffy after a free-kick but the striker failed to catch the dropping ball properly, offering David Button an easy save from eight yards.

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Then Brighton doubled their lead with sweet finesse. Bissouma showed nimble feet to trick his way past Tom Huddlestone at the edge of the area and then aimed a delicate curling shot past the keeper. It bounced out off the inside of the post but Locadia reacted sharply to prod in the rebound from close range.

Derby came back from two goals down to get past Southampton in the previous round but the dullness of their play here made such a transformation seem more unlikely. Lampard tried to spark something, nonetheless. He said he gave a strong talk at half-time to “shake the players out of their passiveness”. He also made a double substitution, introducing Cole and Jack Marriott.

The result was a much improved showing in the second period. As Derby committed more men forward to moderately interesting effect, Brighton filed back, defending competently while remaining on the lookout for opportunities to counterattack. Knockaert led a charge down the left in the 70th minute, concluding the break with a low curling shot from 20 yards that Roos tipped behind for a corner.

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Everything was going rather swimmingly for the home side until Cole stepped forward to make his latest impact on the FA Cup in the 81st minute. Following a move began by a smart through-ball from Huddlestone, the still-lithe veteran showed fresh reflexes and expert technique to guide a header into the net from 12 yards after Dan Burn sent a rash clearance his way. Derby cranked up the pressure in a frantic finale but it was too little, too late.

“This has been the best lesson for the players,” said Lampard, who hopes the education will help his team climb back into the Championship’s promotion places. “We have 15 games left in the league and if our performances are like the second half here, then we have a real opportunity. If they are like the first half, then we don’t.”