Trailing 2-0 to a brace from James Rodriguez in Rio de Janeiro, Uruguay threw everything they had at Colombia in the final 40 minutes but with Ospina inspired in goals, Jose Pekerman's Colombians advanced to their maiden quarter-final at a FIFA World Cup.
With Uruguay having not put a shot on target before half-time in the round of 16 clash, Ospina was made to work much harder in the second half, completing four saves, with a block on Maxi Pereira in the 79th minute particularly telling.
Tabarez claimed the Nice gloveman was the reason his team could not recover.
"We had to take risks to reduce the deficit but we came up against a 'keeper who performed extraordinarily well and didn't let us back into the match," the 67-year-old said.
"But I'm proud of these players. Once again they showed character and gave it their all."
Tabarez also praised Rodriguez for his stunning first-half strike, which put Colombia 1-0 ahead, and conceded Uruguay barely touched the ball in the opening 45 minutes.
In the 28th minute, Rodriguez controlled the ball on the edge of Uruguay's defensive penalty area, swivelled and struck a brilliant volley in off the underside of the bar.
Just five minutes after half-time, Colombia opened up Uruguay's defence with a wonderful cross from Pablo Armero, which Juan Cuadrado nodded down for Rodriguez to notch his fifth goal of the World Cup.
"We knew Colombia had a very good side, one that is able to keep possession and featuring very good players who contribute to the team," he said.
"In the first half, they dictated the tempo and we conceded one of the goals of the tournament. In the second half, we made mistakes and conceded again."
