It was all square at the Stadio Olimpico until 40 seconds into the second half. Goicoechea caught a routine cross from the byline then inexplicably dropped the ball into his net with no other players near him.
Roma coach Zdenek Zeman was already under pressure for a spate of bad results and for keeping Maarten Stekelenburg and Daniele De Rossi on the substitutes bench. The Czech tactician stuck to his guns for this game and fans booed the team off the pitch at the end. The side are still waiting to secure their first league victory of 2013.
Security concerns over Cagliari's IS Arenas stadium handed the Giallorossi a 3-0 win without a ball being kicked back in Week 4.
Erik Lamela returned from suspension, but Federico Balzaretti was injured. Cagliari had managed to keep hold of their prized assets during the transfer window amidst their financial problems.
The Sardinians got off to a dream start after only three minutes. Marco Sau was allowed time and space on the right flank, and, after bamboozling Marquinhos with a series of stepovers, pulled the ball back for Radja Nainggolan to shoot past three static markers into the bottom corner.
Lamela blazed over after Francesco Totti's cushioned header, then saw a shot deflect wide off Luca Rossettini.
The visitors seemed happy to soak up Roma's pressure then hit them on the counter attack. Sau almost doubled the lead on 21 minutes, but dragged his shot wide after Thiago Ribeiro released him on the right hand side.
The Giallorossi equalised ten minutes before the break. Thiago Riberiro fouled Dodo on the edge of the box and Florenzi teed up Totti to smash a low free kick past the wall and into the net. Michael Agazzi got a hand to it, but had no chance of keeping it out.
Roma simply self-destructed in the second half. Goicoechea set the tone when he fumbled Danilo Avelar's innocuous cross from right by the corner flag. The young goalkeeper redeemed himself by saving one-on-one with Sau, but the forward soon netted when he headed Victor Ibarbo's cross into the roof of the net. It was shocking defending once again from Roma, who were all at sea and caught ball watching.
Totti tried to rally his side and stung Agazzi's palms twice in quick succession, before seeing a third chance guided into the side netting.
On 69 minutes, Thiago Ribeiro saw a shot come back off the post. Two minutes later, Sau repeated the trick after curling his shot over the hapless Roma defence and Goicoechea. This time, Francesco Pisano was on hand to tuck the rebound under Goicoechea and into the net.
Lamela's low cross just evaded Pablo Osvaldo, who was sliding in at the far post, as the home side didn't give up the ghost. Marquinho did register a consolation deep into stoppage time. The substitute was given two bites at the cherry to score from close range in the box.
AS Roma: Goicoechea; Piris, Marquinhos, Burdisso, Dodò; Bradley, Tachtsidis, (De Rossi 59) Florenzi; (Marquinho 60) Lamela, Osvaldo, Totti
Cagliari: Agazzi; Pisano, Rossettini, (Perico 77) Astori, Avelar; Ekdal, Conti, (Casarini 81) Nainggolan; Thiago Ribeiro, (Cossu 88) Ibarbo, Sau.
Goals: Nainggolan 3 (C) Totti 35 (R) Goicoechea OG 46 (C) Sau 53 (C) Pisano 71 (C) Marquinho 90+3 (R)